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Transcript

REVIEWS OF MODERN PHYSICS, VOLUME 73, JULY 2001 The physics of manganites: Structure and transport Myron B. Salamon Department of Physics and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 Marcelo Jaime MST-NHMFL, MS E536, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (Published 29 August 2001) The fundamental physical properties of doped LaMnO3, generically termed ‘‘manganites,’’ and much of the underlying physics, were known more than 40 years ago. This article first reviews progress made at that time, the concept of double exchange in particular, and points out the missing elements that have led to a massive resurgence of interest in these and related materials. More recent research is then described, treating first the ground states that emerge as divalent atoms are substituted for trivalent La. A wide range of ground states appear, including ferromagnetic metals, orbital- and charge-ordered antiferromagnets, and more complex stripe and spin-glass states. Because of the interest in so-called colossal magnetoresistance that occurs in the ferromagnetic/metallic composition range, a section is devoted to reviewing the atypical properties of that phase. Next the high-temperature phase is examined, in particular, evidence for the formation of self-trapped small polarons and the importance of Jahn-Teller coupling in this process. The transitions between the high-temperature polaronic phase and the ferromagnetic and charge-ordered states are treated in a fourth section. In each section, the authors stress the competition among charge, spin, and lattice coupling and review the current state of theoretical understanding. They conclude with some comments on the impact that research on these materials has on our understanding of doped oxides and other strongly correlated electronic materials. VII. Implications Acknowledgments References CONTENTS I. Introduction II. Background A. Early experiments B. Early theoretical ideas 1. Double exchange 2. Magnetic and transport properties III. Low-Temperature States A. Parent compounds B. Doped compounds 1. Charge ordering at half filling: n⫽⬁ 2. Charge ordering at half filling: n⫽2 3. Other fractional doping levels 4. Theoretical situation C. Ferromagnetic regime, 3D materials 1. Magnetic properties 2. Electrical transport properties 3. Thermal properties a. Thermal conductivity of 3D materials b. Thermal conductivity of layered materials c. Thermoelectric properties and heat capacity IV. High-Temperature Behavior A. Polaron effects—3D materials B. Polaron effects—layered manganites V. Ferromagnetic/Paramagnetic Phase Transitions A. Theoretical background B. Nature of the phase transition C. Two-phase behavior D. Phase-separation models E. Hall effect in the transition region VI. Charge and Orbital Ordering Transitions A. 3D manganites B. Layered manganites 0034-6861/2001/73(3)/583(46)/$29.20 583 585 585 588 588 589 590 590 592 592 593 594 595 597 597 600 604 604 605 623 625 625 I. INTRODUCTION Colossal magnetoresistance manganites (i.e., based on LaMnO3 and relatives) are known for the unusually large effect an external magnetic field has on their ability to transport electricity and heat. While many of these effects were known more than 50 years ago (van Santen and Jonker, 1950), an appreciation of the size of these effects is a more recent development (Jin, Tiefel et al., 1994). A sensitivity of this magnitude is not observed in any bulk metallic system where, mostly at low temperatures, magnetoresistance arises from a field dependent electronic mean free path. This effect is measurable only in those metals that have a large mean free path in zero field, i.e., very clean systems, where the magnetic field reduces the mean free path by inducing the electrons to move in orbits. Even then, the resistivity change is usually limited to a few percent in practical magnetic fields. Giant-magnetoresistance multilayer metallic films show a relatively large sensitivity to magnetic fields. The mechanism in these films is largely due to what is known as the spin-valve effect between spin polarized metals. If an electron in a regular metal is forced to move across a spin-polarized metallic layer (or between spin-polarized layers) it will suffer spin-dependent scattering. If the electron was initially polarized parallel to that of the layer the scattering rate is relatively low; if originally polarized antiparallel to that of the layer the scattering is high. (The reverse is also possible.) The effect of an ex- 606 608 608 612 613 613 614 616 618 619 620 620 622 583 ©2001 The American Physical Society 584 M. B. Salamon and M. Jaime: Manganites: Structure and transport ternal field is to increase the ratio of the former events, reducing the latter, by aligning the polarization of the magnetic layer along the direction of the external field. This effect is a few tens of percent, and has the very important advantage of not being limited to low temperatures. Spin-valve devices have been used in the magnetic storage industry for several years now, in the form of magnetoresistive read heads. While the physical mechanism that produces the magnetoresistance is well understood, the technological challenges involved in the production of small devices of high sensitivity are the bottleneck of an industry ever hungry for smaller-fasterbetter sensors. Magnetic systems of great potential are those with a limited ability to transport electricity in zero field, resulting from competing dissimilar ground states. In these systems, magnetic fields produce truly dramatic effects by inducing phase transitions or increasing the temperature of already existing phase transitions. Examples of this behavior are found in EuO, pyrochlores, and manganites. Also small gap semiconductors like Ce3Bi4Pt3 and YbB12 can be considered in this category, although a detailed discussion of their ground state is beyond the aim of this review. The limitation of semiconductors in general is that room temperature is a large energy scale. If the gap is large enough to be in the intrinsic conduction limit at room temperature, the magnetic fields required to see changes are unattainable. If the gap is small and magnetic fields within reach, then the effects are limited to very low temperatures (Jaime et al., 2000). In a sense manganites are ideal compounds for magnetic sensor devices, since the two competing ground states are metallic and semiconducting, respectively, and because the energy scale of the phenomenon produces the most interesting effects, i.e., the metal-insulator transition and hence the maximum sensitivity to external fields, at temperatures close to room temperature (Jin, McCormack et al., 1994). See Fig. 1 for recent data on single crystals. It is exactly these features that inspired a tremendous effort from experimentalists and theoreticians to reexamine these materials and to understand the mechanisms involved. Despite this effort neither practical applications nor a satisfactory understanding of the physics of manganites has yet emerged. In the meantime, our perception of the manganites has changed radically. They have been redefined from what seemed a straightforward application in the magnetic-storage industry to a challenge of colossal dimensions from the condensed-matter physics point of view. Soon after the rediscovery of these materials, theoreticians pointed out that the theoretical framework used in the past to explain their behavior qualitatively does not survive when confronted with a quantitative analysis (Millis et al., 1995). With an understanding of the complexity of the problem came the realization of the uniqueness of manganites as a test field for condensedmatter physics theories. Manganites are prototypical of correlated electron systems where spin, charge, and orbital degrees of freedom are at play simultaneously, and where classical simplifications that neglect some interacRev. Mod. Phys., Vol. 73, No. 3, July 2001 FIG. 1. The temperature dependence of xx (H,T) of a La2/3(Pb, Ca)1/3MnO3 single crystal at various fields H. Inset: M H (T) in the transition region. From Chun, Salamon, Lyanda-Geller et al., 2000. tions to study others in detail simply do not work. We are convinced that the comprehension of complex problems posed by the manganites has the potential to produce a substantial advance in the theory of condensed matter, and will eventually shed light into other unsolved problems, for example, high-T c superconductivity. Several excellent review articles (Ramirez, 1997; Coey et al., 1999; Tokura and Tomioka, 1999; Rao et al., 2000) and edited books (Rao and Raveau, 1998; Tokura, 2000) are available in the literature now which discuss the properties of manganites, mostly from a materials science point of view. A review of the optical properties of manganites and related materials has recently appeared (Cooper, 2001). Our focus is different since we choose to emphasize the physics and mechanisms that determine the properties of manganites in different temperature regions. This approach originated in the realization that the most relevant energy scale in these complex systems is the temperature at which the magnetic ordering is observed, and that the interplay of magnetic, charge, and spin degrees of freedom changes qualitatively and quantitatively depending on whether the temperature is much smaller than, similar to, or much larger than the ordering temperature, irrespective of the chemistry. In this review we focus on the present understanding of what we think are the most remarkable physical properties of manganites, observed by state of the art experiments and interpreted with modern theories. We have organized the discussion into three main modules: (i) low temperatures, or energy scales much smaller than that responsible for the metal-insulator, magneticparamagnetic phase transition, (ii) high temperatures, or M. B. Salamon and M. Jaime: Manganites: Structure and transport energy scales much bigger than the energy scale of magnetic interactions, and (iii) intermediate temperatures, or energy scales comparable to the interactions where semiconducting and metallic ground states are both highly unstable and susceptible to such external parameters as temperature, magnetic field, electrical field, crystalline disorder, dimensionality, and doping. Before the discussion of the physical properties we introduce the subject with a brief historical summary of both experimental results and theory developments since the discovery of manganites in the early 1950s. We end the review with a discussion of the implications of manganites. II. BACKGROUND A. Early experiments Manganese compounds of composition AMnO3 (A ⫽La, Ca, Ba, Sr, Pb, Nd, Pr) crystallize in the cubic structure of the perovskite mineral CaTiO3 (Verwey et al., 1950), and will hereafter be referred to as manganites. Depending on the composition they show a variety of magnetic and electric phenomena, including ferromagnetic, antiferromagnetic, charge, and orbital ordering. If the site A is partially occupied by two different atoms, one trivalent as, for example, La and one divalent as Ca, then Mn3⫹ and Mn4⫹ coexist in the samples and the compounds show different behavior as the temperature is changed. The different phase transitions that the mixed manganites display (metal-insulator; charge, orbital, and spin degree of freedom; order-disorder transitions) are sensitive to external parameters such as the pressure and magnetic field. The large sensitivity of the transport properties to external magnetic fields has been optimized recently, and because of this manganites are also known as colossal magnetoresistance (CMR) oxides (Jin et al., 1994). Some manganese compounds crystallize in quasi-two-dimensional (2D) structures (Ganguly, 1984), called layered manganites. Their physical properties, also related to the presence of both Mn3⫹ and Mn4⫹ ions placed in the center of an oxygen octahedron in the samples as in the case of the 3D compounds, are in addition anisotropic because the MnO2 planes, where the magnetic correlations and electrical conductivity take place, are isolated by two AO planes (Moritomo et al., 1996). Due to these properties manganites and layered manganites have attracted a great deal of attention during the last years. Manganites belong to the group of highly correlated systems where charge, spin, and lattice degrees of freedom are intimately interrelated and have the potential to help us improve our understanding of complex systems. The study of the properties of manganites started 50 years ago, soon after Jonker and van Santen discovered a striking correlation between the Curie temperature (T C ), saturation magnetization (M S ), and electric resistivity () in samples of La1⫺x A x⬘ MnO3, where A ⬘ ⫽Ca2⫹, Sr2⫹, and Ba2⫹, when measured as a function of x (Jonker and van Santen, 1950; van Santen and Jonker, Rev. Mod. Phys., Vol. 73, No. 3, July 2001 585 1950). Their polycrystalline samples of composition max La0.7Sr0.3MnO3, Fig. 2, showed maximum values T C max ⬇370 K, M S ⬇90 G/g, corresponding to full polarization of all 3d electrons present in the sample, and electrical conductivity max⬇300 ⍀ ⫺1 cm⫺1, comparable to single-crystal samples grown more recently (Tokura et al., 1994). A significant research effort followed soon after which included the study of the cobaltites (Jonker and van Santen, 1953) and low-temperature measurements in manganites such as the specific heat, magnetization, dc and ac resistivity, magnetoresistance, magnetostriction, I-V curves, dielectric constant, Seebeck effect, and Hall effect (Volger, 1954; Jonker, 1954). The conclusions arrived at by these early researchers were quite close to those reached by careful examination of results in recent experiments, a surprising fact considering the time elapsed, the progress in sample preparation techniques (direct consequence of more than a decade of high-temperature superconductivity research), and the advances in our understanding of condensed-matter physics. Some of the most relevant conclusions reached by researchers in the 1950s are the following: (a) After studying the correlations between crystal structure and the Curie temperature and finding that different samples with the same lattice constant had different Curie temperatures they concluded that a picture of simple exchange interaction could not explain the ferromagnetic transition temperature in manganites (Jonker and van Santen, 1950). We now know that the relevant parameter in determining T C is not the distance between manganese, but the angle of the Mn-O-Mn bond. This is often characterized by the tolerance factor f⫽( 具 r A 典 ⫹r O)/ 关 &(r Mn⫹r O) 兴 , which compares the Mn-O separation with the separation of oxygen atom and A-site occupant. These distances are approximated by the ionic radii of the constituent atoms, suitably averaged and f⫽1 for spherical atoms packed in the perovskite structure. (b) They found that samples of composition x⬇0.3 showed maximum Curie temperature and minimum electrical resistivity, and uncovered a linear relationship between the magnetoresistance and the magnetization of the specimens, concluding that magnetism and electrical conductivity were definitively correlated, Fig. 3 (Volger, 1954). (c) They established that both divalent element content and oxygen stoichiometry determined the Mn4⫹ content in the samples. (d) Their alternating current measurements showed frequency dependence and a magnetoresistance that decreased with the applied voltage. These results clearly pointed to an inhomogeneous phase, and they proposed a model of metallic grains surrounded by a high-resistance intergrain material. We believe today that inhomogeneity is intrinsic to the manganites and may play a very important role in their physics. 586 M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 2. Magnetic properties of La1⫺x Srx MnO3 vs composition: (a) Curie temperature ⌰ and (b) saturation magnetization I S at 90 K vs Sr content in percent. (c) Resistivity vs Sr content x. Crosses in (a) and (b) indicate nominal compositions. From Jonker and van Santen, 1950, and van Santen and Jonker, 1950. From Seebeck effect measurements they concluded that the ferromagnetic ordering had a rigorous influence upon the band structure. The Hall effect proved extremely difficult to observe, implying extremely low mobility (mean free path of the order of interatomic distances), and was anomalous in sign. They attributed these to mixed conductivity, but we know today that they are also the signature of charge localization and small polaron conduction. FIG. 3. Early data on the dc magnetoresistance ⌬/ of La0.7Sr0.3MnO3 as a function of the magnetization at (a) room temperature and (b) 77 K. From Volger, 1954. Not long after those pioneering experiments, Wollan and Koehler (1955) published an extensive neutron diffraction study of the series La1⫺x Cax MnO3, identified the type of magnetic order displayed by the end compounds (Fig. 4), and built the first magnetic structurebased phase diagram for the manganites. This early phase diagram closely matches our modern versions (Schiffer et al., 1995). Almost simultaneously (Jonker, 1956) we find the first attempts in the literature to correlate the crystalline structure with the magnetic properties from the point of view of a then-new magnetic interaction proposed by Zener (1951), the so-called double exchange interaction. According to this picture, in the configuration Mn3⫹ –O2⫺ –Mn4⫹, the easy simultaneous transfer of an electron from Mn3⫹ to O2⫺ to Mn4⫹ causes at the same time a high electrical conductivity and, by the tendency of the traveling electron to retain its spin orientation, a parallel orientation of the mag- (e) (f) Rev. Mod. Phys., Vol. 73, No. 3, July 2001 M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 4. Difference between neutron-scattering intensity at 4.2 K and room temperature: (a) LaMnO3 showing type-A ordering (inset); (b) CaMnO3, indicating type-G order. From Wollan and Koehler, 1955. netic moments of the Mn3⫹ and Mn4⫹ ions. Jonker also found that as the structure of the samples approaches a stable cubic perovskite defined by a tolerance factor f close to unity (Jonker and van Santen, 1950), the saturation magnetization of the specimen approaches the spin only contribution of all Mn ions. Jonker came just short of concluding what these data cry for, i.e., closer to cubic structure⇒closer to colinear Mn3⫹ –O2⫺ –Mn4⫹ bonds⇒stronger double exchange interaction. Further progress came somewhat later when the group at Manitoba accomplished the first growth of high quality millimeter long single-crystal manganites of composition (La, Pb)MnO3 (Morrish et al., 1969). In a series of five papers they confirmed the results previously obtained in ceramics and improved the understanding of the physics (Leung et al., 1969; Searle and Wang, 1969). By that time the non-Heisenberg nature of the ferromagnetic transition was established and a phenomenological model based on a strongly spin-polarized conduction band was proposed to explain it, with the results Rev. Mod. Phys., Vol. 73, No. 3, July 2001 587 FIG. 5. Comparison of magnetization and resistivity models with experiment: (a) Reduced magnetization M(T)/M(0) as a function of the reduced temperature T/T C . The experimental points are taken from Leung et al., 1969. The dashed curve is calculated in the molecular-field approximation, while the solid curve is calculated with the Searle and Wang model. (b) Similar comparison for the normalized resistivity vs T/T C . The solid curve is calculated using the experimental magnetization. From Searle and Wang, 1969. shown in Fig. 5(b) (Searle and Wang, 1969). The one thing that is missing in this whole period is the addition of the electron-phonon interaction. Even though the lattice was clearly linked to the occurrence of ferromagnetism and the metallic phase (Jonker, 1956), a direct attempt to find changes in the rhombohedral Bragg angles at the Curie temperature (Oretzki and Gaunt, 1970) seemed to indicate little lattice involvement. Today we know that hints for the lattice involvement are in fact there, clearly shown by experimental results that reveal the local structure around Mn ions (Louca et al., 1997). During the 1970s little progress was made in the understanding of the manganites, as attention was focused on studies of other interesting magnetic systems like the Eu chalcogenides (Esaki et al., 1967; von Molnar and 588 M. B. Salamon and M. Jaime: Manganites: Structure and transport Kasuya, 1968) and EuO (Penney et al., 1972). The extensive study of these compounds drove the development of the concept of localization of charge by magnetic polarons, magnetically ordered clusters originated by exchange between bound carriers, and localized spins (Kasuya and Yanase, 1968; Kasuya et al., 1970; Kasuya, 1970a, 1970b). Other key developments include the work by Mott and Davis (1971) on amorphous conductors and charge localization by crystalline defects. Dormant manganites reappear in the 1980s in a study by a Japanese team (Tanaka et al., 1982), where oxygen stoichiometry and transport properties were revisited. In spite of the huge effort and remarkable theoretical advances of the 1970s a convincing explanation for the behavior of manganites remained elusive. Tanaka et al. could not arrive at a unique interpretation of their results, in part because of what resulted in a second failed attempt to measure the Hall effect. They proposed for the first time a plausible interpretation based on charge localization into a small magnetic or lattice polaron that conducts by hopping in the paramagnetic region above T C , but the mechanism producing such localization was still unknown. They found the resistivity to have a thermally activated form and the Seebeck coefficient to vary inversely with temperature. The activation energy and the energy scale associated with the Seebeck coefficient differ significantly which, as we note below, is a signature of polaronic behavior. The 3D manganites are the end (n⫽⬁) members of the Ruddlesden-Popper series (A, A ⬘ ) n⫹1 Mnn O3n⫹1 (Ruddlesden and Popper, 1958). Interest in the properties of small n members was also triggered by the discovery of the behavior of the n⫽⬁ compounds described previously (Gorter, 1963; Bouloux et al., 1981). Similar to the n⫽⬁ case, intense and systematic research was renewed only after the discovery of large magnetoresistance (Moritomo et al., 1996; Battle, 1996) in the n⫽2 member. In these compounds the MnO6 octahedra are arranged in n planes, separated by two (Ln, A)O layers. Electrical conduction, magnetic correlation, and orbital ordering take place in the MnO2 planes, and their number and separation determine physical properties. Even though many of the experimental data have been available for a long time, one cannot help but notice some lack of perspective in the early interpretations, characterized by the emphasis on only certain of the physical mechanisms. In order to improve the understanding of the manganites a much broader point of view is necessary, one that is able to put together the whole body of experimental data, including precise crystallographic details, in addition to the double exchange ideas of Zener (1951), Anderson and Hasegawa (1955), Goodenough (1955), and de Gennes (1960). The magnetic and lattice polaron ideas of Kasuya (1959), Kasuya and Yanase (1968), and Mott and Davis (1971), the developments in the theory of small lattice polarons by Holstein (1959), the narrow-band model of Kubo and Ohata (1972), and the concept of Jahn-Teller distortion of the oxygen octahedra surrounding each Mn ion Rev. Mod. Phys., Vol. 73, No. 3, July 2001 (Reinen, 1979) are essential ingredients. To these we must add ideas about phase segregation and intrinsic inhomogeneous behavior explored recently by Moreo et al. (1999). In what follows we shall try to summarize the present situation and discuss why the understanding of the physics of manganites is still incomplete and challenging. B. Early theoretical ideas 1. Double exchange Soon after Jonker and Van Santen (1950; van Santen and Jonker, 1950) discovered the strong correlation between ferromagnetism and metallic conductivity in doped manganites, Zener (1951) offered an explanation that remains at the core of our understanding of magnetic oxides. Zener noted that, in doped manganese oxides, the two configurations 1 :Mn3⫹O2⫺Mn4⫹ and 2 :Mn4⫹O2⫺Mn3⫹ (1) are degenerate and are connected by the so-called double-exchange matrix element. This matrix element arises via the transfer of an electron from Mn3⫹ to the central O2⫺ simultaneous with transfer from O2⫺ to Mn4⫹. Zener points out that the degeneracy of 1 and 2 , a consequence of the two valencies of the Mn ions, makes this process fundamentally different from conventional superexchange. Because of strong Hund’s coupling, the transfer-matrix element has finite value only when the core spins of the Mn ions are aligned ferromagnetically, again distinguished from superexchange which favors antiferromagnetism. As usual, the coupling of degenerate states lifts the degeneracy, and the system resonates between 1 and 2 if the core spins are parallel, leading to a ferromagnetic, conducting ground state. Zener estimates the splitting of the degenerate levels to be given by the ferromagnetic transition temperature k B T C and, using classical arguments, predicts the electrical conductivity to be ⬇ xe 2 T C , ah T (2) where a is the Mn-Mn distance and x, the Mn4⫹ fraction. This provided a qualitative description of the data then available. Anderson and Hasegawa (1955) revisited Zener’s argument, treating the core spin of each Mn ion classically, but the mobile electron quantum mechanically. Designating the intra-atomic (Hund’s) exchange energy by J and the transfer matrix element by b, Anderson and Hasegawa found that Zener’s level splitting is proportional to cos(/2), where is the classical angle between the core spins. The fundamental result, which has remained a cornerstone of double exchange theory, is that the effective transfer integral becomes t eff⫽b cos(/2). The energy is lower when the itinerant electron’s spin is parallel to the total spin of the Mn cores. They also show that the assumption of classical spins can be avoided if one replaces cos(/2) by (S 0 ⫹1/2)/(2S⫹1), where S 0 is M. B. Salamon and M. Jaime: Manganites: Structure and transport the total spin of the two Mn ions and the mobile electron and S is the core spin. Anderson and Hasegawa made the interesting conjecture that a plot of the inverse susceptibility versus temperature would exhibit pure Curie-law behavior at high temperatures (effective T C ⫽0), curving downward as the actual ferromagnetic transition is approached, lying between that of an ordinary ferromagnet and a ferrimagnet. As we will see, the observed behavior is somewhat more complicated. After a gap of some years, de Gennes (1960) revisited this problem, treating the effect of double exchange in the presence of an antiferromagnetic background. He considered a layered material with N magnetic ions per unit volume, each spin S coupled ferromagnetically to its z ⬘ neighbors on the same layer with exchange energy J ⬘ and antiferromagnetically to z neighbors on adjacent layers with energy J. When the angle between the magnetization of successive layers is ⌰, the exchange energy can be written as E ex /N⫽⫺z ⬘ J ⬘ S 2 ⫹zJS 2 cos ⌰. (3) The double exchange contribution is calculated in the tight-binding approximation, using the AndersonHasegawa effective transfer integral, to be E m /N⫽⫺xb ⬘ ␥ k⬘ ⫹b ␥ k cos ⌰/2, (4) where ␥ k and ␥ k⬘ are, as usual, the sum of phase factors over the z and z ⬘ nearest neighbors, and x is the Mn4⫹ concentration. The ferromagnetic state is stable at all temperatures only if the concentration x⭓4JS 2 /b. At lower concentrations, there is a low-temperature transition to a canted state that extends to x⫽0; the canting angle between successive planes is given by ⌰ 0 ⫽2 cos⫺1(bx/4JS 2 ), reaching as x→0. When x ⬍2.5JS 2 /b, the canted state gives way to antiferromagnetic alignment along a critical line, while for 2.5JS 2 /b ⭐x⭐4JS 2 /b the intermediate state is ferromagnetic. The boundary between ferromagnetic and antiferromagnetic intermediate states occurs when the canting angle is ⌰ 0 ⫽2 cos⫺1(2.5/4)⫽103°. In a concluding section, de Gennes anticipates current research by considering selftrapping of a carrier by distortion of the spin lattice, an entity we would now refer to as a magnetic polaron. He demonstrates that at small values of x, local distortions of the antiferromagnetic structure always tend to trap the doped-in charge carrier. In a remarkably complete treatment, Kubo and Ohata (1972) considered a fully quantum mechanical version of a double exchange magnet. They introduced the nowstandard Hamiltonian H⫽⫺J 兺 i, , ⬘ 共 Si • , ⬘ 兲 c i† c i ⬘ ⫹ 兺 i,j, t ij c i† c j , (5) where c i† , and c i are creation and annihilation operators for an e g electron with spin on a Mn site and t ij is the transfer-matrix element. The spin due to t 2g electrons is Si ; is the Pauli matrix, and J is the intraatomic exchange energy, typically referred to as the Hund’s-rule exchange energy. Kubo and Ohata argue Rev. Mod. Phys., Vol. 73, No. 3, July 2001 589 FIG. 6. Reduced magnetization M/M 0 vs the reduced temperature T/T C : 䊊, for La0.62Pb0.38MnO3; 䉭, La0.69Pb0.31MnO3; dashed curve, unmodified, reduced molecular-field curve for S⫽1.81; solid curve, modified curve for a quartic term coefficient of 0.125. From Oretzki and Gaunt, 1970. that it is preferable to calculate the motion of a spindown hole moving in a background of spin S⫹1/2 ions. They transform to a hole version of the Hamiltonian and use a projection-operator method to consider only the lowest-energy Hund’s-rule states. Consider two sites containing a single hole. If both core spins are up, the hole spin must be down, giving a total spin of 2S⫺1/2. On the other hand, if the core spins are oppositely aligned, the configuration belongs to a manifold of total spin 1/2. The corresponding effective transfer-matrix element ranges between 4t/5 for the aligned case and t/5 in the antiparallel configuration for t ij ⫽t for next neighbors. These may be considered to be the average values of cos /2 for the quantum version of the problem. The net result is that the width of the down-spin hole band broadens in the ferromagnetic limit, while that of the up-spin hole band narrows, leading to the expected half metallic ferromagnetic ground state (Irkhin and Katsnel’son, 1994), in which the Fermi surface lies entirely within one spin subband. 2. Magnetic and transport properties As a result of the change in bandwidth, the Curie temperature increases as the system magnetizes. Kubo and Ohata (1972) carried out a mean-field calculation that takes into account the change in effective Curie temperature, finding that the magnetization increases more rapidly than it would if it followed the usual Brillouin function. Figure 5(a) compares the calculated magnetization with that for a conventional mean-field model with S⫽2; the experimental data shown in the plot are from Leung et al. (1969) Similar data for other materials are shown in Fig. 6. 590 M. B. Salamon and M. Jaime: Manganites: Structure and transport In addition to the band structure and magnetic properties of the double exchange model, Kubo and Ohata considered the spin-wave spectrum and the resistivity in the low-temperature limit. The double exchange mechanism broadens the down-spin hole band, the bottom of which is at an energy W/2 below the (narrow) up-spin band, where W is the T⫽0 width of the down-spin band. If the hole band is filled to a Fermi level ⑀ F , down-spin states are ⫽W/2⫺ ⑀ F below any up-spin state. As a consequence, processes that involve the emission or absorption of a single magnon, and therefore scatter a down-spin hole into an up-spin state, are suppressed exponentially as exp关⫺(/kBT)兴. Kubo and Ohata considered two-magnon processes that do not require a spin flip, and predicted that the resistivity will vary as T 9/2 at low temperatures. As we shall see, the T 2 temperature dependence expected for single-magnon processes is exponentially suppressed at low temperatures in double exchange magnets. Polaron transport. More or less concurrent with the development of double exchange theory, Holstein (1959) and collaborators studied the properties of charge carriers that are ‘‘clothed’’ in a distortion of the embedding crystal lattice: the polaron problem. When the hopping probability is large, the charge carrier hops each time a neighboring site acquires, via thermal motion, the necessary lattice distortion. In this case, the polaron is said to be in the adiabatic limit. A characteristic property is that the electrical conductivity is activated and has the form 共 T 兲 ⬀T ⫺1 exp共 ⫺E /k B T 兲 , (6) where E is approximately half of the polaron binding energy. The thermoelectric power, as in a semiconductor, is inversely proportional to temperature, but the characteristic energy is much smaller than E . The simple explanation is that the polaron carries with it only its own chemical potential, but not the energy associated with its accompanying lattice distortion. Much discussion has centered on the existence of a Hall effect in the hopping regime (Entin-Wohlman et al., 1995). The ‘‘drift velocity’’ in this case is the hopping distance divided by the time between successive hops (dwell time). This cannot be construed as a velocity with which to calculate a Lorentz force. Rather, as Holstein and collaborators (Friedman and Holstein, 1963; Emin and Holstein, 1969) demonstrated, the Hall effect arises through the interference effects induced by the Aharonov-Bohm phase produced when magnetic flux threads the loops defined by distinct hopping paths between two sites. The smallest such loop involves one intermediate site and always gives rise to a negative Hall coefficient (Holstein, 1973). The resulting Hall coefficient is also activated but, when the elemental hopping trajectory involves three sites, has an activation energy E H ⬇2E /3. These characteristics of polaron transport were noted above, and will be discussed below, as determinants of polaron-dominated transport. A particular aspect of the electron-lattice coupling in the manganites is the existence of a singly occupied e g Rev. Mod. Phys., Vol. 73, No. 3, July 2001 level in Mn3⫹ and its absence in Mn4⫹. The doubly degenerate e g level is split if the local symmetry is lower than octahedral, and such a distortion will be expected to occur, lowering the energy of the occupied state, but at a cost in lattice energy. Such effects are generally termed Jahn-Teller distortions (Ashcroft and Mermin, 1976). Indeed as we shall see, the crystal structure of the parent compound LaMnO3 is orthorhombic as a consequence of a collective distortion of the structure and gives rise to a Jahn-Teller splitting of the e g orbitals by 1.5 eV (Satpathy et al., 1996b). An Mn4⫹ site, however, does not contribute to this lattice distortion, and should be ‘‘more cubic’’ than its Mn3⫹ neighbors. It is generally agreed that the net energy gain due to the Jahn-Teller distortion can bind a charge carrier to its site, creating a small polaron. Many authors refer to the charge carrier in the activated regime as a ‘‘Jahn-Teller polaron.’’ III. LOW-TEMPERATURE STATES A major motivation for the study of the manganites is the richness of the low-temperature phases that emerge upon doping. The end members of the series, RMnO3 and DMnO3 , where R is a rare-earth atom and D is a divalent substituent such as Ca or Sr, are antiferromagnetic insulators. Mixtures of the two exhibit various magnetic and charge-ordered ground states, which are reviewed in this section. A. Parent compounds Two of the end-members of the series, LaMnO3 and CaMnO3 , were studied in some detail by Wollan and Koehler (1955) more than 40 years ago. The subject has recently been revisited by Huang et al. (1997) in a detailed neutron scattering study of LaMnO3 . The x-raydiffraction work of Yakel (1955) showed that stoichiometric LaMnO3 is orthorhombic, belonging to space group Pnma, while CaMnO3 is cubic, space group Pm3m. The magnetic ground state of LaMnO3 is antiferromagnetic and was labeled A type by Wollan and Koehler. As confirmed by later workers, the magnetic moments lie in the a-c plane and are ferromagnetically aligned along the a axis. Successive planes along b are aligned antiparallel, as sketched in the inset of Fig. 4(a). In this orthorhombic setting of the crystal, the a and c axes are along Mn-Mn directions, with a⫽11.439 Å and c⫽11.072 Å (Yakel, 1955). It is also convenient to use a monoclinic cell with a m ⫽c m ⫽7.960 Å because of its close correspondence to the underlying cubic perovskite cell. CaMnO3 also orders antiferromagnetically, and may be considered as two interpenetrating face-centered lattices with opposite spin. This C-type order, in the language of Wollan and Koehler, is sketched in Fig. 4(b). The magnetic phases of LaMnO3 and CaMnO3 were explained by Goodenough (1955) in terms of covalent bonding between oxygen and Mn3⫹ and Mn4⫹ ions, respectively. In the case of LaMnO3 , the degeneracy of the single e g orbital favors a cooperative Jahn-Teller dis- M. B. Salamon and M. Jaime: Manganites: Structure and transport 591 FIG. 7. Orbital structures at x⫽0 as a function of the antiferromagnetic interaction J S between the t 2g spins. The energy parameters are chosen to be ˜␣ ⫽70 and ˜ ⫽2.5 [case (A)]. The lower sketches demonstrate how the x 2 ⫺z 2 and y 2 ⫺z 2 orbitals are related to more common representations of d orbitals. From Maezono et al., 1998b. tortion (Kanamori, 1960), and the appearance of orbital order. The issue of orbital and spin ordering has been examined more recently by Maezono et al. (1998b). For the undoped case (all sites occupied by Mn3⫹ ions), Maezono et al. obtain a mean-field phase diagram for spin/ orbital ordering as a function of the antiferromagnetic interaction J s between t 2g core spins. The sequence of phases is sketched in Fig. 7. For intermediate values of J s the A-type antiferromagnetic phase is stabilized with what is termed G-type orbital order. Linear combinations of e g orbitals in this state are in the form d z 2 ⫺x 2 and d z 2 ⫺y 2 , with lobes along Mn-O-Mn directions; i.e., the diagonals of the orthorhombic basal plane. The orbital state alternates between neighboring Mn3⫹ ions; the relative orientation of the spin and orbital order are shown in Fig. 8. Inoue and Maekawa (1995) have suggested, recalling earlier work by de Gennes, that the ground state at low doping would exhibit spiral magnetic order. Mishra et al. (1997), to the contrary, have demonstrated that canted (and presumably spiral) states depend on electronic parameters and are not necessarily the ground state of a double exchange model. More recently, an ab initio density-functional calculation (Popovic and Satpathy, 2000) has shown that the orbital and structural order arise simultaneously, driven by bandstructure energy contributions. Detection of orbital ordering requires sensitivity to the asphericity of the electronic charge density in the orbitally ordered state, which leads to intensity at Bragg peaks indexed as (h00) and (0k0), with h,k odd, that are nominally extinct. In this case, Murakami, Hill et al. (1998) observed resonant intensity at the (300) reflection, peaking at an x-ray energy 3 eV above the manganese K absorption edge. The resonant signal at the K edge arises from splitting ⌬ among unoccupied 4p levels via, perhaps, Coulomb interaction between 4p and polarized 3d levels, as sketched in Fig. 8. With the incident radiation polarized normal to the scattering plane () Rev. Mod. Phys., Vol. 73, No. 3, July 2001 FIG. 8. Schematic view of the orbital and spin ordering in the a-b plane of the perovskite manganite LaMnO3. This is the orbital G spin A state shown in Fig. 7. The schematic energylevel diagram of Mn 4p x,y,z in the orbitally ordered state is shown below. From Murakami, Hill et al., 1998. and with frequency , the cross section is zero unless the scattered x rays are polarized () in the scattering plane, in which case I res, ⬇ ⌬2 A 4 sin2 . 兵 1⫹4 关共 ⫺ 0 兲 /⌫ 兴 2 其 2 ⌫ 2 (7) Here 0 is the resonant energy and ⌫ ⫺1 is the lifetime of the excited state. The angle is zero when the crystal c FIG. 9. Azimuthal-angle dependence of the intensity of the orbital ordering reflection (3, 0, 0) normalized by the fundamental reflections (2,0,0) and (4,0,0) at E⫽6.555 keV and at room temperature. The solid curve is the calculated intensity. Inset: Schematic view of the experimental configuration and definition of the polarization directions. From Murakami, Hill et al., 1998. 592 M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 10. Structures of the n⫽1, n⫽2, and n ⫽⬁ members of the Ruddlesden-Popper series of lanthanum manganites. From Moritomo et al., 1996. axis is normal to the scattering plane and 90° when in the scattering plane. A key test that the observed intensity arises from orbital ordering is found in the dependence of the (300) intensity on , as shown in Fig. 9. Some questions (Benfatto et al., 1999) have been raised as to whether the observed signal is directly attributable to orbital ordering or to Jahn-Teller distortions of the same symmetry. Materials of the type AMnO3 are the end (n⫽⬁) members of so-called Ruddlesden-Popper series A n⫹1 Mnn O3n⫹1 . The n⫽2 members of the series also show dramatic magnetoresistive effects, and have been widely studied. This compound has space group I4/mmm and, as seen in Fig. 10, consists of square MnO planes arranged in pairs, spaced from each other by AO planes, and from adjacent pairs, by double AO planes. The ‘‘parent’’ compound in this series, corresponding to all manganese atoms nominally in the Mn3⫹ state, is R 2 SrMn2O7. Almost all work reported has been done on samples doped to have the formula R 2⫺2x Sr1⫹2x Mn2O7 with x⭓0.3. These will be discussed in the next section. the b and c axes in the parent compound interchanged. The magnetic order is antiferromagnetic of the chargeexchange (CE) type described by Wollan and Koehler (1955). A sketch of the complex orbital, charge, and spin order is shown (Tokura and Tomioka, 1999) in Fig. 11. While the spins are purely antiferromagnetic, the charge and orbital ordering occurs in alternate b-c planes, contrary to expectations for large V NN , giving rise to charge-stacked order. A Monte Carlo simulation carried out recently (Hotta et al., 1999) considered Hund’s rule J H , Jahn-Teller , and antiferromagnetic J AF interactions. Using either a 4⫻4⫻2 bilayer or a 4⫻4⫻4 cube, they find that the observed c axis charge-stacked state has lowest energy at intermediate values of J AF so long as V NN does not exceed ⯝0.2 eV. The bare value V NN ⯝3.6 eV is reduced below this critical value by the large dielectric constant of the manganites. Stabilization of this state requires the assistance of cooperative JahnTeller phonons. At larger and smaller values of J AF , the B. Doped compounds 1. Charge ordering at half filling: n ⫽⬁ Upon doping, the manganites exhibit a wide variety of ordered states, including ferromagnetic and chargeordered phases, in addition to the antiferromagnetism and orbital ordering described above. In those compounds in which the proportions of Mn3⫹ and Mn4⫹ ions are rational fractions, charge- and orbital-ordering effects are particularly pronounced. Charge ordering might be expected as a consequence of nearest-neighbor Coulomb repulsion V NN . However, the observed ordered structure in the 3D materials suggests a more complicated explanation. The first clear evidence for charge ordering was found in La0.5Ca0.5MnO3 by Chen and Cheong using electron microscopy (Chen and Cheong, 1996). Electron-diffraction images exhibit commensurate superlattice peaks at low temperature, consistent with alternation of Mn3⫹ and Mn4⫹, as expected for dominance of Coulomb repulsion. The crystal structure of this phase can be indexed in space group Pbnm, with Rev. Mod. Phys., Vol. 73, No. 3, July 2001 FIG. 11. Spin, charge, and orbital ordering pattern of the CE antiferromagnetic type observed for most of the x⬇1/2 manganites. The e g -orbital ordering on Mn3⫹ sites is also shown. The Mn4⫹ sites are indicated by closed circles. Note that Mn3⫹ sites form chains along a, contrary to expectations from Coulomb considerations. From Tokura and Tomioka, 1999. M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 12. Charge/orbital-ordered phase diagrams of various RE1/2Sr1/2MnO3 and RE1/2Ca1/2MnO3 (RE⫽Pr, Nd, and Sm) plotted in the magnetic field-temperature plane. The phase boundaries have been determined by measurements of the magnetic-field dependence of resistivity and magnetization at fixed temperatures. Low-temperature data were obtained utilizing pulsed magnetic fields up to 40 T. From Tokura and Tomioka, 1999. stable states of the system are G-type antiferromagnetic (neighboring Mn spins antiparallel) and ferromagnetic states, respectively. This close relationship among various phases is observed experimentally. Substitution of Nd for La narrows the bandwidth and destabilizes the ferromagnetic state. Nd0.5Sr0.5MnO3 exhibits a narrow CE-type, charge ordered state (Kajimoto et al., 1999), while Pr0.5Sr0.5MnO3 , with a still narrower band exhibits A-type antiferromagnetic order and remains conductive (Kawano et al., 1997). Indicative of the delicate balance among charge, spin, and orbital ordering, Pr0.5Ca0.5MnO3 is reported to order in the chargeordered, CE antiferromagnetic phase (Tokura and Tomioka, 1999). Another manifestation of this balance is found in the field-induced melting of the charge ordered state. This was first reported by Kuwahara et al. (1995) for Nd0.5Sr0.5MnO3 and subsequently to occur in Pr-Ca, Nd-Ca (Tokunaga et al., 1998), Pr-Sr (Tomioka et al., 1995), and Sm-Ca analogs, as seen in Fig. 12. The transition is, as seen, highly hysteretic and accompanied by drop in resistivity by as much as four orders of magni- 593 tude at low temperatures, as seen in Fig. 13. The sequence of phases with various combinations of trivalent and divalent ions in equal proportion has been reviewed in detail recently by Tokura and Tomioka (1999). A related compound, La0.5Sr1.5MnO4 , similar to the 214 family of cuprate superconductors and having equal numbers of Mn3⫹ and Mn4⫹ ions, was found by Moritomo et al. (1995) and Bao et al. (1996), using transport and electron diffraction, to exhibit charge and magnetic order. These results were confirmed by neutron diffraction (Sternlieb et al., 1996) and by x-ray diffraction at the (3/2, 3/2, 0) superlattice position, making use of anomalous dispersion terms for Mn3⫹ and Mn4⫹ ions at the K edge (Murakami et al., 1998). The latter result was part of a path-breaking experiment by Murakami et al. in which orbital ordering, which doubles the charge ordered cell, was detected. The combined magnetic, orbital, and charge structure is shown in Fig. 14. Due to the anisotropy of the e g wave functions, the atomic scattering factors must be treated as tensors near an absorption edge. These give measurable intensity at the K edge at the otherwise forbidden (3/4, 3/4, 0) position. The intensity varies with azimuthal angle as in La0.5Ca0.5MnO3 and shows the required change in polarization. This work represented one of the first direct observations of orbital ordering, although earlier reports of (1/4, 1/4, 0) reflections in electron diffraction (Moritomo et al., 1995; Bao et al., 1996) may also have been due to orbital ordering. 2. Charge ordering at half filling: n ⫽2 In the n⫽2 bilayer compounds, equal numbers of Mn3⫹ and Mn4⫹ ions are expected for the composition RSr2Mn2O7. Here, however, charge ordering is less prevalent. Using electron diffraction, Li et al. (1997) and Kimura et al. (1998) detected the presence of charge order in LaSr2Mn2O7 at 110 K via an increase in resistivity and the appearance of superlattice peaks in x-ray diffraction. The additional peaks have the wave vector q ⫽(1/4,1/4,0), and are consistent with charge/orbital ordering. However, recent neutron-diffraction results (Kubota, Fujioka et al., 1999) find that it does not persist FIG. 13. Changes in the resistivity of a Nd1/2Sr1/2MnO3 crystal after cooling to T⫽2.49 K in zero field. The resistivity shows jumps with increasing and decreasing fields at lower and upper critical fields, respectively, as indicated by crosses. From Kuwahara et al., 1995. Rev. Mod. Phys., Vol. 73, No. 3, July 2001 594 M. B. Salamon and M. Jaime: Manganites: Structure and transport tution of La for Nd reduces this lower Néel temperature, leaving antiferromagnetic order of the Mn ions in which the Mn bilayers form ferromagnetic sheets with the moments aligned and antialigned with Mn-O-Mn bond directions. Below the Néel temperature T N ⬇150 K of the Mn spins in NdSr2Mn2O7 , Maezono et al. (2000) predicted ordering of d x 2 ⫺y 2 orbitals concurrent with A-type antiferromagnetic order. A remarkable study by Takata et al. (1999) combined Rietveld analysis of powder-diffraction data with the maximum-entropy method. In this method, all deformations of the electron density allowed by symmetry are considered, and result in electron density maps which, as shown in Fig. 15, clearly demonstrate the ordering of orbitals into the expected d x 2 ⫺y 2 configuration. 3. Other fractional doping levels FIG. 14. Schematic view of the charge, spin, and orbital ordering in a layered perovskite manganite, La0.5Sr1.5MnO4. The stacking vector along the c axis, shown in the figure, means that the second layer is shifted by one unit cell in this direction. From Murakami, Kawada et al., 1998. at 80 K. They indicate that the magnetic order is predominantly A type, with some admixture of CE antiferromagnetic order and suggest a competition between orbital order and charge localized d x 2 ⫺y 2 d 3x 2 ⫺r 2 /d 3y 2 ⫺r 2 charge order. This result is not surprising, as the Monte Carlo results (Hotta et al., 1999) for bilayers suggest that the charge ordered state is stable only over a very narrow part of the phase diagram without cooperative Jahn-Teller distortions. The series from R⫽Nd to R⫽La has been studied in some detail by Moritomo and co-workers (1999). At low temperatures (T⭐20 K) the Nd compound has been found to order in a complicated antiferromagnetic state, in which both Mn and Nd moments are canted (Battle et al., 1996). Substi- It is natural to expect that charge ordering phenomena might appear at other rational values of doping level. Indeed, Cheong and Chen (1998) have suggested that carrier concentrations of x⫽N/8 per Mn atom characterize special points on the phase diagram. It is somewhat difficult to make this case apart from the x⫽4/8 materials discussed in the previous section. There appears to be a boundary between CE- and C-type antiferromagnetic order in the La1⫺x Cax MnO3 system near x⫽7/8. When Mn3⫹ and Mn4⫹ are in the ratio 1:2, as they are in La1/3Ca2/3MnO3 , the system was thought to order into diagonal charge stripes (Cheong and Hwang, 2000). The stripe order is closely related to the x⫽1/2 charge ordering, with an extra stripe of Mn4⫹ inserted between diagonal Mn3⫹-Mn4⫹ rows. The original interpretation, based on electron diffraction and termed the bistripe mode, has been shown to be inconsistent with more recent x-ray and neutron data (Radaelli et al., 1999). The two structures are compared in Fig. 16. In many of these materials, a ferromagnetic phase appears in a range of compositions, optimized for a wide range of materials when the Mn4⫹/Mn3⫹ ions are in the FIG. 15. Contours containing a charge density of 0.4 e Å⫺3 , obtained by maximum-entropy analysis of synchrotron-x-ray powder-diffraction data for NdSr2Mn2O7 at 19 K, showing the MnO6 double layers. The schematic orbital structure is also shown. From Takata et al., 1999. Rev. Mod. Phys., Vol. 73, No. 3, July 2001 M. B. Salamon and M. Jaime: Manganites: Structure and transport 595 FIG. 16. Schematic diagram of charge ordering patterns in La1/3Ca2/3MnO3 showing alternating chains of Mn4⫹ (open circles) and Mn3⫹ (orbitals) separated by double chains of Mn4⫹ sites (bistripe model). The Wigner-crystal model is in better agreement with observations. From Radaelli et al., 1999. ratio 3:5 (x⫽3/8) (Cheong and Hwang, 2000). It is, of course, this range of compositions that is of the most interest, as these are the most metallic compounds. We defer a discussion of the low-temperature properties of the ferromagnetic composition range to the next section. Figure 17 shows the sequence of phases for Nd1⫺x Srx MnO3. The sequence A-type antiferromagnetic to ferromagnetic to A-type antiferromagnetic to C-type antiferromagnetic is typical of all the 3D perovskites. The intermediate CE phase at x⫽1/2 is not universally present. The surrounding diagrams in the figure sketch the orbital ordering deduced from the x-ray studies described above. The n⫽2 compounds have been studied over a narrower composition range, but also show a variety of lowtemperature phases (Kubota, Fujioka et al., 1999; Argy- FIG. 17. Phase diagram of Nd1⫺x Srx MnO3 crystals. The sketches illustrate the orbital structure types associated with each phase. The letters within the phase diagram refer to paramagnetic (P), ferromagnetic (F), A-type antiferromagnetic (A), CE-type antiferromagnetic (CE), and C-type antiferromagnetic (C) spin ordering. From Okuda et al., 1999. Rev. Mod. Phys., Vol. 73, No. 3, July 2001 riou et al., 1999). In La2⫺2x Sr1⫹2x Mn2O7, the ferromagnetic order below x⫽0.32 has all spins normal to the double MnO planes, designated in Fig. 18 as FMII. Between x⫽0.3 and x⫽0.4, the spins lie in the plane (FM-I). Beyond x⫽0.4, there is canting, or else an admixture of FM-I and AFM-II, with spins in the double planes parallel, but antiparallel stacking of successive double planes. 4. Theoretical situation Since Goodenough’s early work (Goodenough, 1955) on covalence effects, the sequence of phases as a function of doping has remained a topic of considerable interest. Semicovalency, a term coined by Goodenough, arises when the overlap of spin-polarized sp orbitals of manganese ions with unoccupied orbitals of the oxygen allow only covalent bonds involving electrons of one spin direction. Four possible Mn-O-Mn configurations FIG. 18. Magnetic phase diagram of layered La2⫺2x Sr1⫹2x Mn2O7 (0.30⭐x⭐0.50). AFM-I (AFM-II) indicates the planar A-type AFM structure with AFM (FM) intrabilayer coupling and FM (AFM) interbilayer coupling. FM-I and FM-II stand for the ferromagnetic structures with spin within the ab plane and along the c axis, respectively. As for x⫽0.50, only AFM-I exists in phase I, while AFM-I and CEtype AFM coexist in phase II. From Kubota, Fujioka et al., 1999. 596 M. B. Salamon and M. Jaime: Manganites: Structure and transport TABLE I. The sequence of phases as the fraction x of divalent substituents is increased. The magnetic and orbital states are indicated (Goodenough, 1955). x range 0⭐x⭐0.1 0.2⭐x⭐0.4 0.5⭐x⭐0.75 0.75⭐x⭐0.85 0.9⭐x⭐1.0 Type ␣  ␦ ␥ ⑀ Transport insulating conducting insulating insulating insulating Magnetic order A-type antiferromagnetic ferromagnetic CE-type antiferromagnetic C-type antiferromagnetic G-type antiferromagnetic are possible: antiferromagnetic if both empty Mn bonds point toward the bridging oxygen; ferromagnetic if one such bond connects to the oxygen, the other Mn-O bond being essentially ionic; paramagnetic in the absence of covalency; and ferromagnetic and conducting if there is disorder that allows double exchange coupling. In this scheme, the end points can be explained, as noted above. At x⫽1 all six Mn-O-Mn bonds are antiferromagnetic giving rise to G-type order, while at x⫽0, only four bonds can be covalent, leading to A-type antiferromagnetic order. Table I summarizes the sequence of phases predicted by Goodenough. At intermediate concentrations, there is an admixture of adjacent phases. A more recent study of the doping phase diagram has been carried out by Maezono et al. (1998a, 1998b), using a double exchange model in which the orbital degeneracy of the e g levels is included by means of an isospin ជi . The mean-field energy associated with orbital order T ជi 典 T ជi , which is solved along with has the form ⫺  兺 i 具 T spin ordering. For  ⫽0 and reasonable values of the underlying antiferromagnetic superexchange interaction J s , the ferromagnetic state is stable from x⫽0 to x ⬇0.6 at T⫽0. When  is sufficiently large, the model predicts the progression A→F→A→C→G. The CE phase is not found. The phase diagram, along with spin/ orbital ordering patterns, is shown in Fig. 19. The model leading to Fig. 19 does not include Jahn-Teller coupling, and predicts, as seen, that the orbital state at x⫽0 is G-type ordering of y 2 ⫺z 2 and z 2 ⫺x 2 orbitals. This dif- FIG. 19. Free energies for each spin alignment as a function of the antiferromagnetic interaction J S between t 2g spins at x ⫽0. The energy parameters are chosen to be ˜␣ ⫽70 and ˜ ⫽2.5 [case (A)]. These correspond to the orbital structures shown in Fig. 7. From Maezono et al., 1998b. Rev. Mod. Phys., Vol. 73, No. 3, July 2001 Orbital order normal to ferromagnetic sheets disordered charge ordered none fers from the experimentally deduced model shown in Fig. 8. Maezono et al. considered the addition of JahnTeller coupling and find that it tends to suppress the ferromagnetic spin arrangement and to favor the 3x 2 ⫺r 2 /3y 2 ⫺r 2 ordering found experimentally. The calculated ferromagnetic region is less robust than is observed; cf. Fig. 17. Maezono et al. suggest that increasing the bandwidth ⬀t 0 and the antiferromagnetic exchange J s ⬀t o2 , while keeping the other parameters constant, moves the ferromagnetic/antiferromagnetic boundary to larger values of x. Maezono et al. (2000) have also extended their calculations to the layered manganites, including orbital ordering, spin canting, and c-axis lattice deformation. The results are in qualitative agreement with experiment. This model includes on-site Coulomb interactions, but ignores intrasite repulsion. A calculation by Mishra et al. (1997) suggests such terms favor the antiferromagnetic phase near x⫽1/2. A quite different approach has been taken by Yunoki and co-workers (Yunoki, Hu et al., 1998; Yunoki, Moreo et al., 1998) via Monte Carlo simulation. The model includes two e g orbitals, the usual Kondo Hamiltonian for the double exchange model, and Jahn-Teller coupling to classical phonons. The core spins are assumed to be classical. In chains up to 18 sites in length, the carrier density is found to be a discontinuous function of the chemical potential, suggesting phase separation. Extended to clusters of 4 2 sites, the system still exhibits phase separation. Above a critical value of the Jahn-Teller coupling , there is separation between a hole-rich ferromagnetic phase and hole-poor antiferromagnet, as shown in Fig. 20; note that 具 n 典 here is 1⫺x in our notation. This model does not include long-range Coulomb interactions which should prevent macroscopic separation between doped-in holes and their Ca2⫹ or Sr2⫹ donor sites. This point is discussed in a review article by Moreo et al. (1999), where it is suggested that either droplets or polarons may be the most likely configuration. It is not clear the degree to which these 2D Monte Carlo results carry over to 3D manganites. Density-functional methods have been applied to LaMnO3 and CaMnO3 by Satpathy et al. (1996a, 1996b). For LaMnO3 the splitting between t 2g and singly occupied e g levels is found to be 2.0 eV, with a 1.5 eV JahnTeller splitting between d 3z 2 ⫺r 2 (lower) and d x 2 ⫺y 2 orbitals. Both the experimental results and the calculations of Maezono suggest that a linear combination of these is required in the ground state. The bandwidth is found to M. B. Salamon and M. Jaime: Manganites: Structure and transport 597 ones. At low temperatures, the metallic state is characterized by almost fully spin-polarized bands (Park et al., 1998) and a relatively high residual electrical resistivity in the range 50–80 ⍀ cm. 1. Magnetic properties FIG. 20. Monte Carlo simulations of double exchange models with scaled Jahn-Teller interaction parameter : (a) Charge per site 具 n 典 vs chemical potential in the limit that the Hunds’ exchange J H ⫽⬁, using ⫽1.5, antiferromagnetic coupling J ⬘ ⫽0.05; (b) at ⫽1.2 using a 42-site cluster and at T⫽1/50. The two sets of points that produce the hysteresis loops are obtained by increasing and decreasing using ⬃104 sweeps at each ; (c) phase diagram of the 2D model at J H ⫽8.0, J ⬘ ⫽0.05. Jumps that appear in (b) are signatures of phase separation (PS). The phases are denoted S-F and S-AF (FM- and AFM-spin order, respectively) and O-D, O-F, and O-AF (disordered, uniform, and staggered orbital order, respectively). From Yunoki, Moreo, and Dogotto, 1998. be approximately 1.5 eV, with a small gap at the Fermi energy. On doping, the 3z 2 ⫺r 2 band is emptied, leading to the metallic state. Similar results have been reported by Pickett and Singh (1996, 1997a, 1997b). There remain doubts as to which aspects—double exchange, Jahn-Teller effects, polarons, phase separation, among others—are essential to produce a minimal theoretical picture of the manganites. The different contributions are discussed in a recent survey by Loktev and Pogorelov in which they search for common elements needed to produce a strongly field-sensitive metalinsulator transition (Loktov and Pogoretov, 2000). They conclude that the coexistence of various spatial scales, a strong competition between magnetic, orbital, and lattice ordering, and the tendency toward the formation of domains with sharply contrasting properties must remain central factors in any eventual theory. C. Ferromagnetic regime, 3D materials By far, the most attention has been directed to those compositions that have a low-temperature, metallic, ferromagnetic state. Such a phase is a common feature in the phase diagram of the mixed manganites A 1⫺x A x⬘ MnO3, where (A, A ⬘ )⫽(La, Ca), (La, Sr), (La, Ba), (La, Pb), (Sm, Sr), and (Nd, Sr) to mention some of them, for values of x close to 3/8 and extending from 0.2 to 0.5. The metallic state was studied to some extent in the past but not until recently were the intrinsic properties identified and separated from sample-dependent Rev. Mod. Phys., Vol. 73, No. 3, July 2001 Perhaps the simplest question about the ferromagnetic regime in manganites, and one we should be able to answer, is to what extent are manganites standard Heisenberg-like ferromagnets? Magnetization vs magnetic field results in films of (La, Ca)MnO3 reveal typical hysteresis loops with coercivity H c of about 30–50 Oe and a saturation field H S close to 1/2 T (McCormack et al., 1994). The saturation magnetization at T⬇0 matches well the spin only value expected from all 3d electrons present in manganese ions: M S ⫽x⫻Mn3⫹(S⫽4/2)⫹(1⫺x)⫻Mn4⫹(S⫽3/2)⫽4x B ⫹3(1⫺x) B for concentrations x⬇0.3. When doping is lower than 0.2 or larger than 0.5, the saturation magnetization vanishes very quickly, Fig. 2 (Jonker and van Santen, 1950). Measurements in La0.67(Pb, Ca)0.33MnO3 single crystals performed by the authors (Jaime et al., 1998) reveal that the magnetization decreases when the temperature is increased as expected for spin-wave excitations, i.e., M(T)⫽M(0)⫺BT 3/2⫺CT 5/2 ¯ where B⫽0.0587g B (k B /D) 3/2 and D is the spin-wave stiffness constant. At temperatures one order of magnitude lower than the Curie temperature the T 3/2 term dominates the temperature dependence of M, but at T C /2 higher power terms take over, and at T C the magnetization vanishes abruptly deviating from Heisenberg-like behavior, similar to data obtained in Ca-free single crystals [Fig. 5(a)]. The stiffness constant determined from the magnetization measured in a field of 10 kG is D ⫽165 meV Å2 in very good agreement with neutrondiffraction data. Neutron-diffraction studies in the manganites started with the work by Wollan and Koehler (1955) who measured La1⫺x Cax MnO3 for various compositions. They found pure ferromagnetic behavior only in the narrow composition range 0.25⬍x⬍0.4, confirming Jonker’s results on the composition dependent magnitude of the saturation moments. They did not study the temperature dependence in detail. During the last five years, neutron-scattering techniques were exploited to study the manganites in greater detail, and proved to be both extremely interesting and highly nontrivial. At low temperatures and in all compositions, the magnetic inelastic spectra, i.e., energy scans at constant momentum transfer Q, show typically ferromagnetic spin-wave peaks with essentially zero gap and a quadratic dependence on Q at small Q (Lynn et al., 1996; Fernandez-Baca et al., 1998). Well below T C the spin-wave excitations decrease in energy and increase in amplitude when the temperature is increased as expected, Fig. 21. However, as the temperature approaches T C , a new feature develops in the spectrum, consisting of a zero energy quasielastic component also known as the central component. The 598 M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 22. Inelastic neutron-scattering spectra at 250 K (T C ) and Q⫽0.09 Å⫺1 as a function of applied field H. At this temperature there are still spin waves present, but the spectrum is dominated by the quasielastic scattering. In a field of 2 T, the spin-wave signal increases in intensity and sharpens, while the quasielastic response decreases. At 6 T, the spin-wave peaks have moved out of the range of the energy scan. From Lynn et al., 1997. FIG. 21. Magnetic inelastic neutron scattering below T C (a) La1⫺x Cax MnO3 for x⫽1/3 at a momentum transfer Q⫽0.07 Å⫺1. At 200 K the spectrum is dominated by the spin waves in both energy gain and loss. However, as T →T C , a quasielastic (near E⫽0) component develops and grows to dominate the spectrum (Lynn et al., 1997). (b) Similar energy scans for Pr2/3Sr1/3MnO3 (PSMO) and Nd2/3Sr1/3MnO3 (NSMO) at momentum transfer q⫽0.08 (in reciprocal-lattice units). From Fernandez-Baca et al., 1998. central component, more notorious in those compositions with lower T C , persists at least up to 1.25T C with little T dependence above T C and follows the quadratic dependence in Q expected for spin diffusion. The magnetic nature of the central mode has been verified with measurements in magnetic fields. As the magnetic field is increased at constant temperature, the central mode is suppressed and the spin-wave components sharpen as they increase in energy and intensity, Fig. 22 (Lynn et al., 1997). The temperature dependence of the quasielastic Rev. Mod. Phys., Vol. 73, No. 3, July 2001 peak is anomalous. For typical isotropic ferromagnets, such as Ni, Co, Fe any quasielastic scattering below T C is too weak to be observed directly in the raw data (Vasiliu-Doloc et al., 1998), while in some manganites this feature starts to dominate at temperatures well below T C . These results together suggest a correlation between two mechanisms (spin waves on one side and quasielastic neutron-scattering centers, possibly small polarons, on the other) that dominate different temperature, ranges (low and high temperatures, respectively) and coexist in an extended region around T C . We shall discuss both the high-temperature regime and the coexistence region in detail later. The energy dispersion for spin waves has also been measured in detail, and from it the spin-wave stiffness constant was calculated by fitting the low-Q region to the expression E⫽⌬⫹D(T)Q 2 for a number of compounds (Fernandez-Baca et al., 1998). Neutrons do not detect a measurable energy gap ⌬ in the spin-wave dispersion relation; based on the experimental errors an upper limit has been estimated not to exceed 0.05 meV, which indicates that manganites are soft isotropic ferromagnets, comparable to very soft amorphous ferromagnets. As the temperature is increased and the spin-wave energy renormalizes, the stiffness constant decreases following a law expected for two-magnon processes (Mattis, 1981), D 共 T 兲 ⫽D 共 0 兲共 1⫺AT 5/2兲 , (8) M. B. Salamon and M. Jaime: Manganites: Structure and transport 599 FIG. 23. Spin-wave stiffness coefficient D in E⫽E 0 ⫹Dq 2 as a function of temperature: (a) in La0.8Sr0.2MnO3; (b) in La0.7Sr0.3MnO3. (c) in La0.67Ca0.33MnO3 (d) in Nd0.7Sr0.3MnO3 (NSMO) and Pr0.63Sr0.37MnO3 (PSMO). In (a) and (b), the solid curves are fits to the usual Bloch law. D appears to vanish at the ferromagnetic transition temperature, as expected for a conventional ferromagnet; the dashed curves are fits to a power law in (1⫺T/T C ). In (c) D does not vanish at the ferromagnetic transition temperature, in contrast to the behavior of conventional ferromagnets (Lynn et al., 1996). In (d) the spin-wave stiffness coefficient vs T/T C vanishes at T C for PSMO (open circles), but not for NSMO (solid circles). The solid line is the fit to the mode-mode coupling and hydrodynamic theories at high temperatures. The dashed line is an extrapolation to T⫽0 from the low-temperature mode-mode coupling theory. From Fernandez-Baca et al., 1998. where A⫽ 共 0 l 2 /S 兲共 k B /4 D 0 兲 5/2 共 25 兲 , (9) 0 is the unit-cell volume, S is the manganese spin, ( 25 ) is the Riemann zeta integral, and l 2 is a parameter which gives the mean-square range of the exchange interaction, usually comparable to the square of the lattice parameter a 2 . As a rule the stiffness constant decreases when the temperature is increased, as predicted by this formalism, only at low temperatures, and deviates from it as the temperature reaches T C . The temperature dependence in the transition region differs for LaSr and LaCa compounds. For LaSr it is well described by an expression of the form D(T)⬃ 关 (T⫺T C )/T C 兴 ⫺  with the critical exponent ( ⫺  ) very close to the value observed in 3D ferromagnets like iron, cobalt, and nickel (Vasiliu-Doloc et al., 1998). For LaCa compounds, the Rev. Mod. Phys., Vol. 73, No. 3, July 2001 stiffness constant seems to remain finite at T C , a result that clearly goes beyond standard ferromagnetism, as seen in Fig. 23. Computation of the spin-correlation length from neutron measurements of the static wave-vector-dependent susceptibility reveals that while the correlation length diverges at T C ⫽300.9 K for Pr0.63Sr0.37MnO3, it remains finite and around 20 Å down to 0.95T C ⫽0.95⫻197.9 K for Nd0.7Sr0.3MnO3. It has been proposed (FernandezBaca et al., 1998) that magnetism alone cannot explain the exotic spin dynamical properties of these systems and that the increased electron-lattice coupling plays a role. Support for these claims is found in the energy dispersion relation for spin waves. Indeed, when studied in all the way to the zone boundary an anomalous softening/broadening is found in LaCa compounds while the energy dispersion relation for LaSr compounds is 600 M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 24. Magnon dispersion curves open (symbols) along [0,0,] and [,,0] directions for PSMO, NSMO, and LCMO as in Fig. 23 at 10 K. Superposed on these are optical-phonon curves (solid symbols) for LCMO. The solid lines are a fit of a nearest-neighbor spin-wave model to the data for ⬍0.1. The spin waves become strongly damped and nondispersive once their energy intersects the optical-phonon branches. From Dai, Huang et al., 2000. not anomalous. The softening and broadening cannot be explained by the double exchange mechanism; in fact, no correlation between the Curie temperatures and dispersion relation for LaCa, NdSr, and PrSr compounds is found. On the other hand, a remarkable correlation with optical-phonon modes, as shown in Fig. 24, strongly suggests that magnetoelastic coupling (possibly between magnons and Jahn Teller modes) is responsible for the softening (Dai et al., 2000). A theoretical study shows that in fact phonon-magnon interactions can result in magnon broadening in low-T C (narrow bandwidth) manganites provided a large enough magnon-phonon interaction is present (Furukawa, 2000), an interaction that could at least in principle originate in the fact that the metallic state in these compounds is in close proximity to the charge ordered insulating state. In conclusion, La-Sr, and perhaps Pr-Sr, manganite can be well described as a Heisenberg ferromagnet, but other manganites cannot. It may be that in those compounds with smaller tolerance factors and correspondingly lower transition temperatures, large phonon-magnon interactions and Jahn-Teller modes begin to play an important role. 2. Electrical transport properties The low-temperature transport properties of manganites have been studied in some detail only in recent years. Those properties, which require a finite electric current through the sample, have proven to be sensitive to extrinsic effects as grain boundaries and magnetic domain boundaries. These effects are related to grain boundary charge/spin tunneling-limited transport and have attracted a great deal of attention because of the potential applications of manganites as spin-valve-like devices. A large low-field/low-temperature magnetoreRev. Mod. Phys., Vol. 73, No. 3, July 2001 FIG. 25. Two different treatments of the dependence of the resistivity (H,T) on the magnetization M(H,T): (a) an exponential dependence in La0.7Ca0.3MnO3, using data taken at H⫽10, 20, 30, 40, and 50 kOe that progress, respectively, from low to high M values; solid line, a least-squares fit to the data (Hundley et al., 1995); (b) similar analysis for a La1⫺x Srx MnO3 crystal (x⫽0.175). The points are obtained from the (B) and M(B) curves at respective temperatures. The solid line is obtained using the (T) and M(T) data in a field of 0.5 T. The behavior is quadratic at small values of the magnetization. From Tokura and Tomioka, 1999. sistance can be obtained when samples are polycrystals prepared with conveniently small crystallographic grains (Schiffer et al., 1995). This spin-tunneling limited magnetoresistance is based on good spin polarization of individual grains or magnetic layers in the system and vanishes as thermal entropy kills intragrain polarization, more or less rapidly as practical temperatures are reached (Hwang et al., 1996). Epitaxial films do not show low-field magnetoresistance (Li, Gupta et al., 1997). There is also a high-field magnetoresistance component associated with the grain boundaries that is essentially temperature independent between 5 and 280 K, and is most likely related to alignment of spins in a magnetically disordered region near the grain boundaries. The low- and high-field magnetoresistance, studied as functions of the grain size by the group in Barcelona M. B. Salamon and M. Jaime: Manganites: Structure and transport (Balcells et al., 1998), show saturation in low fields and monotonic increase in high fields for grain sizes equal to or smaller than 1 m, revealing the distinct origin of both contributions. La2/3Sr1/3MnO3 samples, where grain sizes are 30 nm or smaller, do not show a metallic electrical resistivity in any temperature range, not even in the low-temperature ferromagnetic phase. This behavior has been attributed to a Coulomb blockade contribution to the resistivity, similar to that observed in such other half metallic systems as CrO2 (Coey et al., 1998). The intrinsic low-temperature transport properties of ferromagnetic manganites are far from trivial, and in fact have proven to be a challenge. The intrinsic magnetoresistance of these compounds vanishes at low temperatures, and correlates well with the magnetization, i.e., saturated magnetization⫽null magnetoresistance. A detailed study of these correlations indicates that the resistivity and magnetization are related by the empirical expression (H,T)⫽ m exp关⫺M(H,T)/M0兴 over a wide temperature range all the way up to the Curie temperature (Hundley et al., 1995), clear evidence of the double exchange interaction at play [Fig. 25(a)]. However, in contradiction to the exponential fit, (H,T) is found to decrease as the square of the magnetization near T C (Tokura and Tomioka, 1999), as seen in Fig. 25(b). As a consequence of the double exchange mechanism, charge carriers at low temperatures are spin-down holes moving in an S⫽5/2 background. There are no propagating up-spin hole states in the S⫽2 manifold; they only exist localized on sites at which the t 2g core is not ferromagnetically aligned. Therefore single-magnon scattering processes, which cause the resistivity of conventional ferromagnets to vary as T 2 , are suppressed. Kubo and Ohata extended the standard perturbation calculation of Mannari (1959) to consider two-magnon processes, predicting a leading T 9/2 temperature dependence of the resistivity. However, a dominant T 2 contribution is universally observed in the manganites, and has usually been ascribed to electron-electron scattering (Urushibara et al., 1995; Schiffer et al., 1995; Snyder et al., 1996). However, there is a general relationship between the coefficient A of the T 2 contribution to the resistivity and the electronic heat-capacity coefficient ␥, pointed out by Kadowaki and Woods (1986). In the manganites, the ratio A/ ␥ 2 is more than an order of magnitude larger (Jaime et al., 1998) than the Kadowaki-Woods ratio for metals in which strong electron-electron interactions are present. This suggests that a different mechanism is involved. In a recent paper (Jaime et al., 1998) resistivity data on single crystals of composition La0.67(Ca, Pb)0.33MnO3 are discussed, demonstrating that the quadratic temperature dependence is rapidly suppressed as the temperature is reduced. They argue that the observed T 2 contribution reflects the reappearance of minority spin states that are accessible to thermally excited magnons. Spin-polarized photoemission data, taken on films exhibiting square hysteresis loops, indicate 100% spin polarization only at low temperatures, decreasing gradually as the temperature is inRev. Mod. Phys., Vol. 73, No. 3, July 2001 601 FIG. 26. Low-temperature behavior of the resistivity: (a) resistivity vs T 2 for magnetic fields up to 70 kG in a thin-film sample of La2/3Ca1/3MnO3 . (b) Numerical derivative of the ⫺1 data, ␣ H / T 2 ; 䊏, for zero field; 䊐, for 70 kG dashed curve, the same quantity calculated for the magnon model with Dq min /kB⫽44 K. Inset: raw data minus calculated residual resistivity (solid squares) and least-squares curve (dotted line) vs temperature. The nonadditivity of the residual resistivity and the quadratic component violate Matthiesson’s rule. From Jaime et al., 1998. creased (Park et al., 1998). Single crystals, which have essentially no hysteresis, would be expected to depolarize more rapidly. Jaime et al. have extended Mannari’s calculation to the situation in which a minimum magnon energy is required to induce spin-flip transitions. At temperatures well below that energy, single magnon scattering is suppressed exponentially as predicted by Kubo and Ohata. The treatment is in the context of the relaxation time approximation while a proper theory would consider lifetime effects from magnon scattering using Furukawa’s many-body approach. Nonetheless, the results are in qualitative agreement with the data. Bandstructure calculations (Pickett and Singh, 1996; Singh and Pickett, 1998) also indicate that minority spin states persist at E F , even at T⫽0 K. Figure 26(a) shows the resistivity of a single-crystal sample of La0.67(Ca, Pb)0.33MnO3 with T C ⫽300 K, vs the 602 M. B. Salamon and M. Jaime: Manganites: Structure and transport square of the temperature in fields up to 70 kOe. The data show a dominant T 2 temperature dependence with evidence of a small T 5 contribution (10 ⍀ cm at 100 K). A calculation of the T 9/2 contribution predicted by Kubo and Ohata for two-magnon processes predicts only 0.5 ⍀ cm at 100 K with appropriate parameters. It is likely, then, that this is the usual T 5 contribution from electron-phonon processes. The inset to Fig. 26(b) shows that the data do not follow a T 2 dependence to the lowest temperatures. Rather, they deviate gradually from the curve 0 ⫹ ␣ (H)T 2 , fit over the range 60⭐T ⭐160 K, before saturating at an experimental residual resistivity exp 0 ⫽91.4 ⍀ cm, comparable to values observed by Urushiba et al. (1995), but ⬃7% larger than 0 . This conclusion is not changed by including the T 5 contribution. Fits to data taken in various fields show that ␣ H decreases with increasing field and is the source of the small negative magnetoresistance at low temperatures. To quantify the disappearance of the T 2 contribution, Jaime et al. numerically differentiate the data, plotting ␣ (H) ⫺1 d /d(T 2 ) in Fig. 26(b). The T 5 contribution, not subtracted, gives a slight upward curvature to the data at higher temperatures. The usual calculation of the electron-magnon resistivity (Mannari, 1959) has been extended to allow the minority-spin subband to be shifted upward in energy such that its Fermi momentum differs by an amount q min from that of the majority subband. This should be a reasonable approximation in the intermediate temperature regime in which both minority and majority bands have substantial densities of states at E F . The one-magnon contribution can then be written as ⑀ (T)⫽ ␣ ⑀ T 2 , where ␣ ⑀⫽ 9 3N 2J 2ប 5 8e 2 E F4 k F 冉 冊 kB m *D 2 I共 ⑀ 兲. (10) In this equation, NJ is the electron-magnon coupling energy which is large and equal to ⫽W⫺E F in the double exchange Hamiltonian of Kubo and Ohata; 2W is the bandwidth. The magnon energy is given by Dq 2 , and we have defined I共 ⑀ 兲⫽ 冕 ⬁ ⑀ x2 dx. sinh2 x (11) 2 2 /2k B T, where Dq min is the The lower limit is ⑀ ⫽Dq min minimum magnon energy that connects up- and downspin bands; this result reproduces Mannari’s calculation in the limit ⑀ →0, and Kubo and Ohata’s exponential cutoff for large ⑀. At high temperatures, the lower limit of the integral in Eq. (10) can be set equal to zero, leaving only the coupling energy NJ⫽W⫺E F as a parameter. Equating the calculated value to the experimental ␣ fixes the coupling to be W⫺E F ⬇1.0 eV or W ⯝1.5 eV, in good agreement with a virtual crystal estimate of the bandwidth (Pickett and Singh, 1997a). Figure 26(b) shows the experimental data (1/␣ H ) /dT 2 , 2 2 2 and ␣ ⫺1 0 d( ␣ ⑀ T )/d(T ) assuming D(0)q min⫽4 meV and including the temperature dependence observed experimentally, D(T)/D(0)⫽(1⫺T/T C ) 0.38 (FernandezBaca et al., 1998) which is important only at higher tem- Rev. Mod. Phys., Vol. 73, No. 3, July 2001 peratures. While the curve follows the data qualitatively, it is clear that the minimum magnon energy is substantially larger than 4 meV at low temperatures, and decreases with increasing temperature. This extension of the magnon resistivity calculation to spin-split parabolic bands greatly oversimplifies the changes in the minority-spin band that accompany magnetic ordering. As a consequence, the calculation cannot be expected to represent accurately the cutoff of magnon scattering due to loss of minority-spin phase space. Nonetheless, the rapid suppression of the T 2 contribution to the resistivity and the agreement between its magnitude at higher temperatures with parameters expected for the manganites confirm the basic picture. In the intermediate temperature regime, 0.2⭐T/T C ⭐0.5 here, the manganites appear to be normal metallic ferromagnets with the resistivity dominated by spin-wave scattering. At lower temperatures, the increasingly half metallic character of the material is manifested by a temperature dependent cutoff of the spin-wave scattering process, leaving in its wake only residual resistivity from the intrinsic doped-in disorder and indistinguishable phonon and two-magnon contributions. As these heavily doped materials have significant disorder and large residual resistivities, we recall that strongly disordered materials also exhibit T 2 regimes below half the Debye temperature (Nagel, 1977). This result, however, an extension of the Ziman theory of liquid metals, sets an upper limit of ␣ T 2 / 0 ⯝0.03 before the resistivity changes to a linear temperature dependence; our ratio is unity at 100 K with no evidence for a linear regime. We conclude that the quadratic temperature dependence is not due to phonon scattering in a strongly disordered material. An alternative explanation of the low-temperature electrical transport is based on an argument by Alexandrov and Bratkovsky that polarons remain the predominant charge carriers even below T c (Alexandrov and Bratkovsky, 1999), and that transport is dominated by polaron tunneling. As a consequence, the lowtemperature resisitivity is predicted to behave as 共 T 兲 ⫽ 0 ⫹E s /sinh2 共 ប s /2k B T 兲 , (12) where E is a constant and s is the softest optical mode. This expression gives a T 2 contribution above T ⫽ប s /k B , which as been determined to be 86 K from fitting to data on polycrystalline La0.8Ca0.2MnO3 (Zhao et al., 2000). However, as we have seen in Fig. 24, the lowest optical mode has an equivalent temperature of 250 K, considerably higher than the polaron picture would indicate. In a recent theoretical effort, Furukawa (2000) has discussed the problem of quasiparticle lifetimes due to magnon scattering processes in half metals, taking into account the effects of spin fluctuations. He argues that there is no minority-spin Fermi surface in a half metal at zero temperature but that spin fluctuations induce a minority band at finite temperatures. At low temperatures, then, the one-magnon scattering self-energy depends on spin fluctuations and increases as T departs from zero. M. B. Salamon and M. Jaime: Manganites: Structure and transport 603 This nonrigid band picture allows unconventional onemagnon scattering processes to arise as the minority band is created and occupied, provided the incoherent limit ␦ m⫽ 关 „M(0)⫺M(T)…/M(0) 兴 Ⰶ1 holds. In this limit the resistivity is expected to be proportional to the cube of the temperature divided by the spin-wave stiffness constant incoherent ⬀(T/D) 3 , where D is defined via q ⫽Dq 2 . As the temperature increases, ␦ m cannot be considered small and conventional magnon scattering is recovered (coherent limit), where the usual T 2 dependence in the resistivity is expected. The crossover temperature is estimated by T * ⯝ 关 2M(0)/0.06W 兴 2 D 3 where W is the bandwidth, M(0) the saturation magnetization, and D the spin-wave stiffness. Reasonable parameters for manganites result in T * ⬇50 K, in good agreement with the temperature where the T 2 regime appears in La0.67(Ca, Pb)0.33MnO3 (Jaime et al., 1998), La1⫺x Srx MnO3 (0.2⬍x⬍0.4) (Furukawa et al., 2000), and bandwidth controlled La0.67(Ca, Sr)0.33MnO3 (Broussard et al., 1999). Similar results have also been observed in a classical half metal, CrO2 (Watts et al., 2000). The Hall effect in the ferromagnetic phase of manganites is particularly puzzling, and a relatively large effort has been made to try to understand it. In ferromagnetic metallic systems, the embedded magnetic moments cause asymmetric scattering of current-carrying electrons, producing a voltage that superimposes on the usual Hall voltage, the so-called anomalous Hall voltage. The total Hall resistance xy can be written as xy 共 B,T 兲 ⫽R H 共 T 兲 B⫹ 0 R S 共 T 兲 M 共 B,T 兲 , (13) where R H (T) is the temperature-dependent Hall coefficient, B is the applied magnetic field (unit demagnetization factor assumed), R S (T) is the temperaturedependent anomalous Hall coefficient, and M(B,T) is the magnetization. Measurements in films of composition La0.67Ca0.33MnO3 (Matl et al., 1998; Jakob et al., 1998) and single crystals of composition La0.67(Ca, Pb)0.33MnO3 (Chun et al., 1999) show an almost temperature-independent Hall coefficient R H and strongly temperature-dependent anomalous Hall coefficient that peaks above the Curie temperature and is not simply related to the ordinary resistivity. Typical results in single-crystal samples are displayed in Figs. 27(a) and (b). Figure 27(a) shows the measured Hall resistance in magnetic fields up to 7 T, for several temperatures below and above T C . At low temperatures (solid symbols) two different regimes (low fields⬅anomalous, high fields ⬅normal) are evident. Figure 27(b) shows in its inset how the two components are separated using the demagnetizing-field-corrected magnetization measured in the same samples. Also in the inset is shown the scaling of the anomalous hall coefficient with the sample resistivity. The body of Fig. 27(b) displays the calculated effective carrier concentration n eff(T)⫽1/eR H , which show a monotonic increase as the temperature is reduced and is as large as 2.4 holes/Mn (almost five times larger than nominal doping level). These large values Rev. Mod. Phys., Vol. 73, No. 3, July 2001 FIG. 27. Hall data taken over a wide range of fields and temperatures: (a) Hall resistivity xy of a La2/3(Pb, Ca)1/3MnO3 single crystal as a function of field; (b) effective number of holes per Mn atom as a function of temperature, calculated from the data of (a), assuming the high-field slope to determine the ordinary coefficient, and n eff⫽e/R0 . The lower inset shows the decomposition of xy into ordinary and anomalous Hall effects below T C . The upper inset shows the linear relation between the anomalous Hall coefficient R S and the longitudinal resistivity xx . From Chun et al., 1999. have been attributed to charge compensation effects. In a two-band model, the Hall coefficient is given by R H ⫽(r h n h h2 ⫺r e n e 2e )/e(n h h ⫹n e e ) 2 and the experimental values in Fig. 27(b) can be reproduced using 604 M. B. Salamon and M. Jaime: Manganites: Structure and transport band-structure calculations (Pickett and Singh, 1997b) and a mobility ratio e /h⫽1.6. The anomalous Hall coefficient and its scaling with the sample magnetization will be discussed in detail later, since it is closely related to the physics of the transition between a hightemperature state where charge carriers are localized as small polarons and the low-temperature ferromagnetic state. Interestingly, layered manganites do not show charge compensation effects and the carrier concentration deduced from Hall-effect measurements in single crystals agrees very well with the values expected from the nominal doping level. The anomalous Hall effect in layered manganites seems to scale with the resistivity, exhibiting a minimum at the same temperature where the electrical conductivity is maximal (Chun, Salamon, Jaime et al., 2000), but more work needs to be done extending the measurements above the Curie temperature to be able to compare them to their 3D counterparts. 3. Thermal properties a. Thermal conductivity of 3D materials The thermal conductivity of ferromagnetic manganites, both in polycrystal (Hejtmanek et al., 1997; Chen et al., 1997; Cohn et al., 1997) and single-crystal samples (Cohn et al., 1997; Visser et al., 1997), is relatively small, with room-temperature values between 1 and 3 W K⫺1 m⫺1 and a low-temperature peak that never exceeds 4 – 5 W K⫺1 m⫺1. These conductivities are 5–10 times smaller than those observed in the normal state of high-temperature superconductor single crystals at similar temperatures (Peacor et al., 1991). In both systems electrons contribute a small (20–40 %) fraction of the total observed thermal conductivity tot in the hightemperature phase. In high-temperature superconductors the mean free path of unpaired electrons increases dramatically below the superconducting transition, causing e to dominate tot somewhat below the transition before dropping at low temperatures. In manganites the transition into the metallic ferromagnetic state shows a similar effect, i.e., the thermal conductivity improves as the electrons (localized in the paramagnetic state) gain mobility in the ferromagnetic phase but e remains a small fraction of tot . Figure 28(a) displays the thermal conductivity in La1⫺x Cax MnO3 polycrystals for different values, Fig. 28(b) shows the thermal conductivity vs temperature in semilog scale for a number of polycrystal and single-crystal samples. In the low-temperature region (the high-temperature paramagnetic region will be discussed later) we notice an appreciable enhancement of the thermal conductivity in ferromagnetic compounds. The Wiedemanz-Franz law can be used to estimate the electronic contribution to the thermal conductivity e in the metallic state. Such an estimate in Cohn’s samples results in an electric contribution no larger than 20% of the total thermal conductivity. A similar estimate for single crystals and films of different Curie temperatures is displayed in Fig. 29(a), where it can be seen that e (T) can only qualitatively explain the lowtemperature increase in . In order to make a quantitaRev. Mod. Phys., Vol. 73, No. 3, July 2001 FIG. 28. Comparison of thermal conductivity for various samples: (a) thermal conductivity vs temperature for La1⫺x Cax MnO3 polycrystals. The inset is a magnified view of the data for x⫽0 and 0.15, showing the anomaly observed near T C ⫽170 K for x⫽0.15. The solid line represents the data for x⫽0 shifted upward by 0.02 W/mK (Cohn et al., 1997). (b) Thermal conductivity for four different manganite perovskite samples. The arrows denote the ferromagnetic transition temperature. The three samples possessing a sharp change in (T) at T C also undergo a metal-insulator transition. The dashed lines represent an exponential temperature dependence ⫽ 0 exp(T/350), with 0 values described in the text. The inset shows the electrical conductivity versus , at 350 K for the four samples. From Visser et al., 1997. tive analysis, we have taken data from Visser’s report and plotted them together with the result of a Wiedeman-Franz estimate and an extrapolation of the high-temperature phonon contribution in Fig. 29(b). Here we can see that e (T) can only account for 40% of the low-temperature value by itself and about 80% of M. B. Salamon and M. Jaime: Manganites: Structure and transport 605 FIG. 29. Separation of thermal conductivity into electron and phonon contributions: (a) Electronic thermal conductivity calculated from electrical resistivity in a La0.67(Ca,Pb)0.33MnO3 single crystal using the Wiedemann-Franz law. (b) Total thermal conductivity exp for La0.2Nd0.4Pb0.4MnO3 single crystal (from Visser et al., 1997), high-temperature phonon contribution ph and electronic contribution e calculated using the electrical resistivity and Wiedemann-Franz law. (c) Temperature dependence of the total thermal conductivity total ⫽ electron ⫹ phonon for a ferromagnetic (T C ⫽205 K) Pr0.6Ca0.2Sr0.14MnO3 single crystal. For the separation of phonon and electron parts of thermal conductivity the electronic contribution was estimated using the experimental resistivity data, corrected for sample porosity and assuming the validity of Wiedemann-Franz law. (d) Similar data for a layered compound. Note different scales for in-plane and out-of-plane curves. From Hejtmanek et al., 1999. the total when added to a simple extrapolation of the high-temperature lattice contribution ph . In Fig. 29(c) we see data for a single crystal of composition Pr0.65Ca0.21Sr0.14MnO3 (Hejtmanek et al., 1999) from which it is clear that an extra contribution is missing from the picture. Two conclusions can be derived from this behavior. In the first place, the lattice thermal conductivity in the ferromagnetic manganites is lower than in the cuprates and is not limited by electron-phonon scattering. This conclusion finds support in the absence of a linear term in the low-temperature electrical conductivity. Consequently we believe that the observed ph is small due to the presence of large cationic disorder and to the scattering of phonons against Jahn-Teller Rev. Mod. Phys., Vol. 73, No. 3, July 2001 modes (an estimate by Cohn et al. found that the phononic mean free path is comparable to the Mn-Mn distance ⬃4 Å). Second, because of the very nature of small magnetoelastic polarons in the paramagnetic phase, as the temperature is reduced and the charge carriers delocalize, the effective electron-phonon scattering rate decreases and both thermal conductivities ( e and ph ) increase in the metallic state. b. Thermal conductivity of layered materials The thermal conductivity of single crystal samples of the layered manganite La1.2Sr1.8Mn2O7 is anisotropic, as expected from the structure and the behavior of other 606 M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 30. The Seebeck coefficient S vs temperature for a mixed Pb-Ca-doped sample. It is positive and metal-like at low temperature, has an anomalous kink near 30 K, and develops a positive field dependence above 40 K. The dashed line is a linear fit in the low-temperature regime. From Jaime et al., 1998. transport properties (Matsukawa et al., 2000). The component along the ab crystallographic direction increases in the ferromagnetic state as in the 3D compounds, while the component along the c axis shows no change. Another contribution to tot in manganites should come from spin waves ( m ), however, an estimate using the specific-heat anomaly ⌬C at T C results in a very small m ⯝0.1 W K⫺1 m⫺1 for polycrystal value, La0.7Ca0.3MnO3 (Cohn et al., 1997). Detailed measurements of the very low-temperature thermal conductivity in good single crystals should tell us more about the relative contribution from spin waves, and could help to better understand the transport properties of manganites. A discussion of the high-temperature thermal conductivity in manganites is presented in Sec. IV. c. Thermoelectric properties and heat capacity The Seebeck coefficient S(T) provides additional information on the nature of transport at low temperatures. Figure 30 shows S(T), measured on the same sample as in Fig. 26 at H⫽0 and 80 kOe. At the lowest temperatures, S(T) is positive, linear in temperature, and extrapolates to zero as T→0. The field dependence is small and negative. The large slope suggests, from the Mott formula, that the resistivity is a strong function of energy at E F . There is a sharp deviation from linear behavior in the temperature range in which the T 2 dependence of the resistivity becomes dominant and the field dependence changes sign and becomes larger. We note that the peak in the low-temperature thermopower that is regularly seen in thin-film samples is absent here, and is therefore not intrinsic to these materials. If we take the scattering to be independent of energy, which is the case below 20 K, the Seebeck coefficient can be ex2 pressed as S(T)⫽( 2 /2e)(k B T/E F ) (Ashcroft and Mermin, 1976). Using the simplistic approximation of a parabolic band E F ⫽ប 2 k F2 /2m * , and spherical Fermi surRev. Mod. Phys., Vol. 73, No. 3, July 2001 FIG. 31. Composition dependence of the heat capacity: (a) From top to bottom, low-temperature specific heat, plotted as C/T vs T 2 , for LaMnO3⫹ ␦ : 䊊, with ␦ ⫽0.26; 䉮, ␦ ⫽0.15; 䉭, ␦ ⫽0.11; for La1⫺x Cax MnO3 䊐, x⫽0.11 〫, x⫽0.33 Ghivelder et al., 1999. (b) Specific heat C for La1⫺x Srx MnO3 , x⫽0.15 under magnetic fields. The inset shows reduction of C upon application of 9 T for x⫽0.15. From Okuda et al., 1998. face k F3 ⫽3n 2 , the effective mass turns out to be m * /m⬇3.7, comparable to the value obtained from specific-heat measurements. The sharp deviation from linear behavior in the temperature range 20–40 K correlates with the onset of electron-magnon scattering which, being a spin-flip process, must involve the minority spin band, and which therefore has a different dependence on energy near E F . Comparable behavior has been observed in CrO2 films (Watts, 2000). The specific heat in the ferromagnetic phase of manganites has received a great deal of attention recently. Measurements at low temperatures in La1⫺x Cax MnO3 have been done in polycrystalline samples (Coey et al., 1995; Hamilton et al., 1996; Ghivelder et al., 1998). Results for x⫽0.1, 0.33, and 0.62 are displayed in Fig. 31(a) between 4 and 10 K. Both experimental groups have attempted to fit the data using three-parameter fits of the form C⫽C e ⫹C l ⫹C m , where C e is the electronic contribution ␥ T, C l is the lattice contribution  T 3 ⫹ ␣ T 5 , and C m is the magnetic M. B. Salamon and M. Jaime: Manganites: Structure and transport contribution, ␦ T 3/2 in the case of a ferromagnet, ␦ 2 T 2 in the case of a standard Néel antiferromagnet. A fit to the experimental results for metallic La0.67Ca0.33MnO3 gives ␥ ⫽4.7 mJ/mol K2,  ⫽0.12 mJ/mol K4, and ␦ ⫽0 indicating the absence of a spin-wave contribution. The electronic density of states at the Fermi energy determined from the measured ␥ is N(E F )⫽3 ␥ /( k B ) 2 ⫽2.0 eV⫺1 Mn⫺1. The lattice contribution indicates a Debye temperature D ⫽430 K. Neutron-scattering measurements (Lynn et al., 1996) provide a direct measurement of the spin-wave stiffness constant D ⫽170 meV Å2 , which would yield a specific-heat magnetic term ␦ ⫽0.62 mJ/mol K5/2, equivalent to 20% of the experimental value at 5 K and above the scatter of the data. The absence of a spin-wave component in the lowtemperature specific heat is not understood at present. On the other hand, measurements in the system La0.7Sr0.3MnO3 do show a spin-wave component, see Fig. 31(b) (Coey et al., 1995; Woodfield et al., 1997). The stiffness constant deduced from it is D⫽130 meV Å2 in rough agreement with the neutron-scattering results D ⫽188 meV Å2 (Martin et al., 1996) and D⫽154 meV Å2 (Smolyaninova et al., 1997), with no indication of a spinwave energy gap. The electronic component in this material is ␥ ⫽3 – 7 mJ/mol K2 implying a density of states at the Fermi energy N(E F )⫽1.3– 2.7 eV⫺1 Mn⫺1, close to the value found in La0.33Ca0.67MnO3 and roughly a factor 2–3 larger than the band-structure estimate (Pickett and Singh, 1996) N(E F )⫽0.8 eV⫺1 Mn⫺1, implying some mass renormalization effect is present. The study of the doping dependence of the Sommerfeld coefficient ␥ in La1⫺x Srx MnO3 (Okuda et al., 1998) reveals a behavior not typical for a filling-controlled metal-insulator transition. Results for La1⫺x Srx TiO3 and La1⫺x Srx MnO3 are displayed in Fig. 32. The critical mass enhancement near the metal-insulator phase boundary (vertical hatching), indicating a canonical Mott transition, in La1⫺x Srx TiO3 is absent in La1⫺x Srx MnO3. A recent heat-capacity study of other compounds reveals more about the manganites behavior. Measurements in Nd0.67Sr0.33MnO3 (T C ⬇200 K) show no spinwave contribution as in the case of La0.33Ca0.67MnO3, but contain a hyperfine contribution ⬃T ⫺2 , a broadened Schottky-like anomaly originating in the Nd spin system and excess entropy attributed to Nd-Mn exchange interaction and responsible for a temperaturedependent ‘‘linear’’ term ␥ (T)T (Gordon et al., 1999). Oxygen deficient samples of LaMnO3 have been measured by the group in Brazil (Ghivelder et al., 1999), in samples with insulating, ferromagnetic, low-temperature phases [Fig. 31(a)]. These compounds, as in the case of La0.7Ca0.3MnO3, do not show a spin-wave contribution. The most surprising feature in both Nd0.67Sr0.33MnO3 and LaMnO3⫺ ␦ is an unexpectedly large zerotemperature linear term, even in the case of insulating compounds. Ghivelder et al. (1999) propose that the spin-wave component and the large electronic component have their origin in ferromagnetic clusters, where mass-enhancement mechanisms (magnetic polarons, latRev. Mod. Phys., Vol. 73, No. 3, July 2001 607 FIG. 32. Doping level (x) dependence of the electronic specific-heat coefficient ␥ for La1⫺x Srx MnO3 compared with that for La1⫺x Srx TiO3 (LaTiO3⫹x/2). The hatched vertical bar for La1⫺x Srx TiO3 (LaTiO3⫹x/2) indicates an antiferromagnetic metallic region sandwiched by AF insulating (x⬍0.05) and PM metallic (x⬎0.08). The hatched region for La1⫺x Srx MnO3 is the metal-insulator transitional region. From Okuda et al., 1998. tice polarons, Jahn-Teller effect, or Coulomb interactions) are at play. It seems strange though that mass enhancement mechanisms have no effect in Ca or Sr compounds, nor in layered manganites as discussed below. Heat-capacity experiments have also been performed in layered manganite La1.3Sr1.7Mn2O7 (Okuda et al., 1999). In this system spin-wave excitations contribute to the specific heat and can be suppressed by a 9-T magnetic field, which results in a 12-K energy gap in the spin-wave spectrum as shown in Fig. 33. The effect is larger than in the case of 3D compounds, but not as large as expected for an ideal 2D system. The authors claim that a finite dispersion of the spin wave along the c axis will explain the results. The zero-field linear term is ␥ ⫽3⫾1 mJ/mol K2, comparable to values in the 3D counterpart La0.7Sr0.3MnO3. The lack of mass enhancement effects in the linear term may imply the presence of an anomalous carrier scattering process or dynamical phase separation (Yunoki, Moreo, and Dagotto, 1998). A relatively large low-temperature magnetoresistance indicates that transport is somewhat limited by electronmagnon scattering, but spin-valve effects and fielddependent localization also play a role. In conclusion, the low-temperature thermal properties of manganites and layered manganites are still far from being completely understood, especially in what concerns spin-wave contributions and mass enhancement effects. More detailed quantitative studies are in order, to test the spin-wave excitation spectrum in those compounds that show the contribution, in single-crystal samples that reveal intrinsic as opposed to sampledependent effects. A summary of heat-capacity results is shown in Table II. 608 M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 33. Heat capacity of layered manganites: (a) temperature dependence of specific heat (C) as shown in a C/T vs T 2 plot in 0 and 9 T for a layered (n⫽2) manganite; solid line the extrapolated C/T value at 0 K, which expresses the ␥ value; (b) reductions of C upon the application of a field of 9 T for La1.3Sr1.7Mn2O7 (LSMO327, x⫽0.35), compared with La0.6Sr0.4MnO3 (LSMOI13, x⫽0.4) along with the calculated results (solid lines) for the simple-square 2D lattice and simple-cubic 3D lattice with an appropriate spin-wave stiffness constant (D⫽190 meV Å2 ). From Okuda, Kimura, and Tokura, 1999. IV. HIGH-TEMPERATURE BEHAVIOR tions become negligible and electric charge/lattice interplay is dominant (Jaime and Salamon, 1999). As noted in Sec. II.B.2, the formation and transport properties of small lattice polarons in strong electronphonon coupled systems, in which charge carriers are susceptible to self-localization in energetically favorable lattice distortions, were first discussed in disordered materials (Holstein, 1959) and later extended to crystals (Mott and Davis, 1971). Emin (1973) and Hillery et al. (1988) considered the nature of lattice polarons in magnetic semiconductors, where magnetic polarons are carriers, self-localized by lattice distortions but also dressed with a magnetic cloud. A transition from large to small polaron occurs as the ferromagnet disorders, successfully explaining the metal-insulator transition observed experimentally in EuO. If the carrier, together with its associated crystalline distortion, is comparable in size to the cell parameter, the object is called a small, or Holstein, polaron. Because a number of sites in the crystal lattice can be energetically equivalent, a band of localized states can form. These energy bands are extremely narrow, and the carrier mobility associated with them is predominant only at very low temperatures. At high temperatures the dominant transport mechanism is thermally activated hopping, with an activated mobility p ⫽ 关 x(x⫺1)ea 2 /h 兴 (T 0 /T) s exp关⫺(WH⫺J3⫺2s)/kBT兴 with a the hopping distance, J the transfer integral, x the polaron concentration, and W H one-half of the polaron formation energy. There are two physical limits for these hopping processes, depending on the magnitude of the optical phonon frequency. If lattice distortions are slow compared to the charge carrier hopping frequencies, the hopping is adiabatic, otherwise it is nonadiabatic. In the adiabatic limit, s⫽1 and k B T 0 ⫽h 0 , where 0 is the optical phonon frequency and in the nonadiabatic limit, we have s⫽3/2 and k B T 0 ⫽(pJ 4 /4W H ) 1/3. The polaronic transport in manganites is usually considered adiabatic, in this case the conductivity is given by ⫽ A. Polaron effects—3D materials In the early experimental study of manganites unexpected high-temperature transport properties were believed to be dominated by nonintrinsic effects like defects, crystalline disorder, grain boundaries, and impurities. Years later, with the preparation of good quality films by laser ablation on lattice-matched substrates and the growth of large single crystals, it became evident that the observed behavior is intrinsic and due to localization of charge carriers in small polarons (Ohtaki et al., 1995; Jaime, Salamon, Pettit et al., 1996; Jaime et al., 1997; Palstra et al., 1997; Chun et al., 1999). The localization is a consequence of a large electronphonon interaction, enhanced by the Jahn-Teller activity of Mn3⫹ in the manganites, and has an impact on the electric and thermal transport properties as well as on the lattice properties. We will see in what follows that the high-temperature regime is where magnetic correlaRev. Mod. Phys., Vol. 73, No. 3, July 2001 ⫽ 冉 冊 ⑀ 0 ⫹W H ⫺J x 共 1⫺x 兲 e 2 T 0 exp ⫺ បaT k BT 冉 0T 0 E exp ⫺ . T k BT 冊 (14) The temperature dependence observed in the hightemperature resistivity of manganites follows this adiabatic prediction very well, as seen in Fig. 34, from temperatures close to T C up to 1200 K (Jaime, Salamon, Rubinstein et al., 1996; Worledge et al., 1996; De Teresa et al., 1998). At high enough temperature magnetic correlations can be completely ignored, since charge/lattice and charge/charge interactions dominate. In this regime on-site Coulomb repulsion (Worledge et al., 1998) has been observed in support of the small polaron picture. Small grain polycrystalline samples, very thin and unannealed films, on the other hand, have been reported to show variable-range-hopping-type localization (Coey et al., 1995; Ziese and Srinitiwarawong, 1998) and nonadiabatic small polaron transport (Jakob, Westerburg et al., 1998). 609 M. B. Salamon and M. Jaime: Manganites: Structure and transport TABLE II. Results from fits to the low-temperature specific heat in manganite compounds. Units are mJ/mol K2 for ␥, K for D , meV Å2 for D sw , mJ/mol K3 for ␦ 2 , and J K/mol for A. The cases where a contribution from spin waves is observed, but its magnitude not reported, are indicated by *. The cases where a spin wave contribution is not observed are indicated by -. Composition Order/T C ␥ D D sw ␦2 A sample Reference SC SC SC SC SC SC SC SC PCC PCC PCC PCC Okuda et al., 1998 Okuda et al., 1998 Okuda et al., 1998 Okuda et al., 1998 Okuda et al., 1998 Okuda et al., 1998 Okuda et al., 1998 Okuda et al., 1998 Woodfield et al., 1997 Woodfield et al., 1997 Woodfield et al., 1997 Coey et al., 1995 La0.85Sr0.15MnO3 La0.84Sr0.16MnO3 La0.83Sr0.17MnO3 La0.82Sr0.18MnO3 La0.80Sr0.20MnO3 La0.75Sr0.25MnO3 La0.70Sr0.30MnO3 La0.60Sr0.40MnO3 La0.90Sr0.10MnO3 La0.80Sr0.20MnO3 La0.70Sr0.30MnO3 La0.70Sr0.30MnO3 FM/230 FM/250 FM/260 FM/290 FM FM FM FM FM FM/320 FM/375 FM/370 0 1.5 4.0 5.2 4.5 4.0 3.5 3.7 0 3.34 3.29 6.0 386 390 388 395 398 420 450 438 255 270 285 353 * * * * * * * * * * 130 - La0.67Ba0.33MnO3 La0.67Ba0.33MnO3 La0.70Ba0.30MnO3 FM/330 FM/340 FM/220 4.4 5.6 6.1 383 416 333 - PCC PCC PCC Hamilton et al., 1996 Hamilton et al., 1996 Coey et al., 1995 La0.90Ca0.10MnO3 La0.67Ca0.33MnO3 La0.38Ca0.62MnO3 La0.53Ca0.47MnO3 La0.50Ca0.50MnO3 La0.47Ca0.53MnO3 La0.80Ca0.20MnO3 La0.70Ca0.30MnO3 FMI/140 FM/250 A-AFM FM/255 CO-AFM FM-AFM FM/250 FM/260 0 4.7 0 8 0 0 7.8 5.2 368 430 490 40 162 390 - PCC PCC PCC PCC PCC PCC PCC PCC Ghivelder et al., 1998 Ghivelder et al., 1998 Ghivelder et al., 1998 Smolyaninova et al., 1997 Smolyaninova et al., 1997 Smolyaninova et al., 1997 Hamilton et al., 1996 Coey et al., 1995 LaMnO3 LaMnO3.11 LaMnO3.15 LaMnO3.26 AFM FM-CG/154 FM-CG/142 SG/48 0 23 19 0 302 369 387 498 55 75 32 16.1 PCC PCC PCC PCC Woodfield et al., 1997 Ghivelder et al., 1999 Ghivelder et al., 1999 Ghivelder et al., 1999 Nd0.67Sr0.33MnO3 FM/280 25 PCC Gordon et al., 1999 Pr0.60Ca0.40MnO3 Pr0.6Ca0.3Sr0.1MnO3 Pr0.6Ca0.2Sr0.2MnO3 Pr0.6Ca0.1Sr0.3MnO3 Pr0.6Sr0.4MnO3 CO-AFM/230-160 FM/196 FM/235 FM/262 FM/305 28.0 59.1 57.34 40.82 33.3 PCC PCC PCC PCC PCC Lees Lees Lees Lees Lees Pr0.50Ca0.50MnO3 Pr0.55Ca0.45MnO3 Pr0.65Ca0.35MnO3 Pr0.65Ca0.35MnO3 (H⫽8.5 T) Pr0.7Ca0.3MnO3 La2.3Ca0.7Mn2O7 CO-AFM/230-180 CO-AFM/230-170 CO-AFM/230-160 CO-FM/200-140 CO-AFM/230-140 23.7 24.2 20.7 7.6 12.2 PCC PCC PCC PCC PCC PCC Smolyaninova Smolyaninova Smolyaninova Smolyaninova Smolyaninova Raychaudhuri La1.3Sr1.7Mn2O7 FM/120 SC Okuda et al., 1999 0 4.72 4.51 5.77 5.40 3 冉 冊 kB ES ⫹b . e k BT (15) Contrary to the case of band semiconductors, where Rev. Mod. Phys., Vol. 73, No. 3, July 2001 1.3 3.80 1.89 - 2.4 3.1 15.7 7.0 30.6 41 The thermopower of small polaronic systems is similar to that of band semiconductors, governed by thermal activation of carriers across a small barrier and thus a function of the inverse temperature: S⫽ 433 344 342 333 339 9.20 9.30 8.69 190 8.03 et et et et et al., al., al., al., al., 1999 1999 1999 1999 1999 et et et et et et al., al., al., al., al., al., 2000 2000 2000 2000 2000 1998 the chemical potential determines the temperature dependence of both the conductivity and the thermoelectric properties and ⑀ 0 ⫽E ⫽E S , small polaronic conduction results in very different characteristic energies, i.e., E S ⫽ ⑀ 0 ⰆE because E includes W H , which implies that for the charge carrier to be able to hop from one site to another, a lattice distortion of energy 2W H needs to be provided. In a band semiconductor the strongest 610 M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 34. The resistivity of an n⫽⬁ manganite, following the adiabatic, small-polaron hopping model at all stages of annealing. Each line is the cooling curve after being held at T h for 10 h. T h ⫽650 K for the top curve, and increases by 50 K for each successive run, up to T h ⫽1200 K for the bottom curve. From Worledge et al., 1996. temperature dependence comes from the availability of carriers in the conduction band; in a small polaron system the limiting factor is the number of available sites that the charge carriers can hop to. For the manganites high-temperature transport measurements reveal E S ⬇ few meVⰆE ⬇100– 200 meV, the signature of small polarons (Fig. 35). The independent term k B b/e in the thermopower is given by the configurational entropy of, in the case of the manganites, placing a hole with spin 3/2 (S h ⫽3/2) moving in a spin-2 background (S b ⫽2), namely, ⫺(k B /e)ln兵关2Sh⫹1兴/关2Sb⫹1兴其⫽⫺(kB /e)ln(4/5) ⫽⫺19 V/K plus the mixing entropy term that counts in how many different ways x holes can be distributed between n Mn sites. The mixing term in the case of correlated 1D hoping with weak near-neighbor repulsion is given by ln关x(1⫺x)/(1⫺2x)2兴 (Chaikin and Beni, 1976). Alternative models were discussed by Heikes (1965), giving in the correlated limit ln关(1⫹x)/(1⫺x)兴 and in the uncorrelated limit (where double occupancy is allowed) ln关x/(1⫺x)兴. None of these models can explain quantitatively the experimental results, and only the uncorrelated limit by Heikes predicts the right qualitative dependence on the carrier concentration x. There are three proposed explanations for this behavior. The first is a disproportionation model, where two Mn3⫹ atoms generate Mn2⫹ and Mn4⫹ sites with the transference of one electron. The disproportionation density is related to oxygen nonstoichiometry (Hundley and Neumeier, 1997). Another possibility is that small polarons move by hopping between divalent atoms in real space due to the elastic stress introduced in the lattice by atomic size mismatch (Jaime et al., 1997). This is a type of impurity conduction where the number of available sites for hopping increases with the carrier concentration x and as a consequence the mixing entropy remains unchanged. Finally, the Heikes uncorrelated limit also suggests that multiple occupancy or collective behavior is possible for small polarons. One of the most distinctive properties of small poRev. Mod. Phys., Vol. 73, No. 3, July 2001 FIG. 35. The resistivity in the adiabatic limit and thermopower vs 1000/temperature. Lines are fits in the high-temperature regime, and E ⫽ ⑀ 0 ⫹W H and E S ⫽ ⑀ 0 . As in Fig. 6, there is a large difference in these two energy scales. From Jaime, Salamon, Pettit et al., 1996. laronic transport is the magnitude and temperature dependence of its Hall mobility H (T). The Hall mobility is not a power law of the temperature as in band semiconductors but thermally activated, with an activation energy E calculated to be always less that for the drift mobility E d . The simplest model predicts E ⬇E /3, experimentally observed in oxygen-deficient LiNbO3 (Nagels, 1980). As first pointed out by Friedman and Holstein (Friedman and Holstein, 1963) the Hall effect in hopping conduction arises from interference effects of nearest-neighbor hops along paths that define an Aharonov-Bohm loop. When the loops involve an odd number of sites sign anomalies arise in the Hall effect and the Hall coefficient is activated with a characteristic energy E H ⫽2E /3: 0 R H ⫽R H exp where 冉 冊 2E , 3k B T (16) 再 g H F 共 兩 J 兩 /k B T 兲 0 ⫽ RH exp ⫺ gd ne ⑀ 0⫹ 冎 4 兩 J 兩 ⫺E S 3 . k BT (17) At sufficiently high temperatures, the contribution from the anomalous Hall effect is sufficiently small to allow a measurement of the polaronic contribution. Figure 36 (Jaime et al., 1997) shows both the sign anomaly (negative Hall effect for hole transport) and activated M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 36. The magnitude of the Hall coefficient ⫺R H vs temperature showing the values obtained by 䉱, ramping the magnetic field up and 䉲, ramping; down in a thin-film sample; 䊏, from a single-crystal sample with a higher transition temperature; dashed line, Arrhenius fit Inset: The natural logarithm of the averaged Hall coefficient for the thin film. The solid line is a linear fit giving 91 meV for the activation energy. From Jaime et al., 1997. behavior of the Hall coefficient expected from the Friedman-Holstein picture in a film sample of (La0.75Gd0.25)2/3Ca1/3MnO3. Indeed, the energy characterizing the exponential rise of the Hall coefficient, E H ⫽91⫾5 meV is about 2/3 the measured conductivity activation energy, E H /E ⫽0.64⫾0.03 in excellent agreement with theory. 0 In order to compare the prefactor R H with the experimental results some assumptions need to be made. The geometrical factor g d depends on the ratio of the probability P nnn of next-near-neighbor (nnn) hops to P nn , that of near-neighbor (nn) hops, through g d ⫽(1 ⫹4P nnn /P nn ). If these probabilities are comparable, i.e., if diagonal hopping is allowed in the Mn square sublattice, g d ⫽5, g H ⫽5/2, and the exponential factor in Eq. (17) becomes exp关(ES⫺兩J兩)/3k B T 兴 ⯝1. In the regime 兩 J 兩 ⭓k B T, the function F( 兩 J 兩 /k B T) is relatively constant 0 ⫽⫺0.02/ne⫽⫺3.8 with a value ⬇0.2, leading to R H ⫻10⫺11 m3/C. This yields an estimated carrier density n ⫽3.3⫻1027 m⫺3, quite close to the nominal level of 5.6 ⫻1027 m⫺3. The main effect of Gd, trivalent as La, is to reduce the Curie temperature by further reducing the Mn-O-Mn bond angle (and hence reducing the bandwidth) without changing the doping level. This trick made the Hall effect measurement possible to temperatures as high as four times T C allowing the observation of the small polaron regime for the first time. Chun et al. have confirmed the picture with recent measurements in slightly underdoped single crystals and show (Fig. 37) how the small polaron picture breaks down as the Curie temperature is approached from above, a remarkable accomplishment given the small size of the samples and the intrinsic experimental difficulties (Chun, Salamon, Tomioka et al., 2000). Note that diagonal hopping is not disallowed by symmetry considerations in manganites because the MnRev. Mod. Phys., Vol. 73, No. 3, July 2001 611 FIG. 37. The activated behavior of Hall coefficient R H (main panel) and Hall mobility H (inset) above T C ⫽220 K for a single crystal of La0.7Ca0.3MnO3. Above 300 K, the slopes are in accord with small-polaron hopping theory. Also shown in the main panel is the effective activation energy of the conductivity, E * . The Hall and resistivity data deviate from polaronic behavior at the same temperature. From Chun, Salamon, Tomioka et al., 2000. O-Mn bond angle is smaller than 180° (regardless of Gd content). Diagonal hopping successfully explains the sign anomaly in the Hall effect, which requires small polarons traversing Hall-effect loops with odd number of legs, predicts correctly the carrier concentration in the sample and also reduces the value of the hopping attempt frequency required to fit the electrical conductivity prefactor (Jaime et al., 1997). Worledge et al. studied the electrical resistivity of (Gd free) La1⫺x Cax MnO3 films in the full composition range 0⭐x⭐1 (Worledge et al., 1998). They find that a model for small polarons that hop adiabatically between nn sites and experience Coulomb repulsion reproduces well the results but underestimates the total conductivity, and propose that considering hopping to more than nearest neighbors (nnn) could fix the problem. Interestingly, a theoretical study based on ab initio density-functional calculations, showing how cooperative Jahn-Teller coupling between individual MnO6 centers in manganites leads to the simultaneous ordering of the octahedral distortion and the electronic orbitals, has found that a nnn hoping integral V ⬘ ⫽⫺0.42 eV comparable to the nn hoping integral V ⫽⫺0.52 eV together with a small but finite third neighbor hopping V ⬙ are necessary to fit the e g bands in the linear muffin-tin orbital theory (Popovic and Satpathy, 2000). The thermal transport of manganites in the paramagnetic state is particularly puzzling because, besides its magnitude being very close to the theoretical minimum value (Cohn, 2000) [see Fig. 28(a)], it does not show the 612 M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 38. The inverse of the dc susceptibility for La0.67Sr0.33MnO3: 䊊, experimental data. Solid and dotted lines are fits to the constant-coupling and Curie-Weiss models, respectively. From Causa et al., 1998. characteristic 1/T temperature dependence for heat carried by phonons with a mean free path limited by anharmonic decay (Berman, 1976). Instead (T) increases with temperature. An exponential increase with equal temperature parameters, found in samples with very different compositions, has been interpreted as due to dynamical lattice distortions, a consequence of T-dependent Debye-Waller factors (Visser et al., 1997). These results however have not been reproduced by other groups (Hejtmanek et al., 1999; Cohn, 2000) and a definitive explanation is still missing. One possibility is that the very moderate increase in with temperature is due to the increase of a relatively small electronic component added to a saturated and temperatureindependent phonon component. Such an increase of the electronic component with temperature could be due to the increase in thermally activated small polaron mobility with temperature. Indeed, in Fig. 29(b) we can see a small but distinguishable increase in the hightemperature electronic component e calculated using the Wiedemann-Franz relation. Electron-spin-resonance (ESR) studies in the paramagnetic phase reveal the nature and interaction properties of Mn3⫹-Mn4⫹ spin pairs. High-temperature ESR experiments were performed by Causa et al. (1998) together with magnetic susceptibility experiments up to 1200 K in La0.67Ca0.33MnO3, La0.67Sr0.33MnO3, Pr0.67Sr0.33MnO3, and La0.67Pb0.33MnO3 . dc (T) follows, in all cases, a ferromagnetic Curie-Weiss temperature dependence dc (T)⫽C/(T⫺⌰), at high enough temperatures (T⬎2⌰). For temperatures below 1.5⌰ the curve for the inverse susceptibility vs T (Fig. 38) shows a positive curvature suggesting spin clustering effects. In the paramagnetic regime the ESR spectrum consists of a single line, with g values that go from 1.992 for Ca compound to 2.00 for Pr-Sr compound, with a linewidth that deviates from linearity and a tendency to saturation with the temperature (Fig. 39). The computation for ESR performed by Causa et al. demonstrates that both ESR and dc have the same temperature dependence clearly indicating that all the Mn ions contribute to the obRev. Mod. Phys., Vol. 73, No. 3, July 2001 FIG. 39. Peak-to-peak magnetic resonance linewidth ⌬H pp vs temperature, showing a universal behavior for the perovskites ⬘ MnO3 with A ⬘ ⫽Ca, Sr, and Pb. Different symbols La0.67A 0.33 refer to X-, L-, and Q-band data. The inset shows similar data for the Pr-Sr analog. From Causa et al., 1998. served ESR spectra and that the ESR linewidth should be related to the relaxation mechanism of the coupled magnetic system. The authors also claim that the temperature dependence of the ESR linewidth may be described by a universal curve, whose temperature scale is associated with T C . The behavior above T C is determined solely by the temperature dependence of ESR (T) and the infinite temperature linewidth kept as an adjustable parameter. No evidence is found of a spinphonon contribution to the experimental linewidth in this regime. These results agree with previously discussed transport experiments that show evidence for decoupling of small polarons from spins above T⫽2T C . Additional evidence for polaron formation is found from structural and optical studies. Pair-distribution analysis of neutron powder-diffraction data, the width of EXAFS features, and Raman data all find evidence for structural features consistent with polaron formation well above T c . A surprising aspect of these results is that such evidence persists well into the ferromagnetic regime. We will return to the question of admixtures of polaronic and metallic phases in Sec. V. Perhaps the strongest evidence that polaron effects are important is found in the isotope effect. Substitution of 16O by 18O lowers the transition temperature by 21 K for La0.8Ca0.2MnO3 (Zhao et al., 1996) and lesser amounts for x⫽0.3 (Babushkina et al., 1999). The shift is attributed to the mass dependence of the polaron bandwidth, and provides strong evidence that oxygen motions play a central role in controlling the ferromagnetic transition. B. Polaron effects—layered manganites The paramagnetic state of layered manganites [small n members of the Ruddlesden-Popper series (La, A) n⫹1 Mnn O3n⫹1 ] has not received as much attention as the corresponding state in 3D compounds. There are a few reasons for this. In the first place, single-crystal samples are difficult to grow and multiple phases are 613 M. B. Salamon and M. Jaime: Manganites: Structure and transport common (Berger et al., 2000); second, the thermal and electric transport properties are strongly anisotropic and results obtained in ceramic polycrystalline samples mix up conductivity tensor components. Because of the structural similarities between 3D and layered manganites (see Fig. 10) it is expected that charge-lattice correlations play an important role in both of them. However, the tetragonal crystal field in n⫽2 compounds can split the degenerate Mn e g orbitals, a condition that precludes the Jahn-Teller effect as the mechanism that drives small polaron formation in the paramagnetic insulator phase of the n⫽⬁ materials. Nonetheless, measurements in single-crystal samples of La1.2Sr1.8Mn2O7 (Zhou et al., 1998) support the existence of charge localization effects due to small polaron formation. Anisotropic behavior is found in the electrical resistivity and thermopower that indicate conduction by Holstein small polarons in the c crystallographic direction (perpendicular to the La-Sr-Mn-O planes) and by ‘‘Zener-pair’’ polarons in the ab crystallographic direction (along La-SrMn-O planes) that condense into clusters as the temperature is reduced. In a recent report, Liu et al. show that electrical transport in polycrystalline samples of La1.4Sr1.6Mn2O7⫹ ␦ also behave as predicted by the small polaron Eqs. (14) and (15), with a thermal activation energy in the electrical conductivity close to 100 meV and a characteristic energy in the thermopower roughly 15–20 times smaller (Liu et al., 2000). These numbers are quite close to those found in the 3D compounds, implying similar small polaron formation energies W H are involved. The authors also report that N2 annealing of the samples reduces the sample resistivity and thermopower and increases the Curie temperature and magnetoresistance. Similar results are reported by Jung in polycrystalline samples of La1.6Ca1.4Mn2O7 where E ⯝100 meV is substantially larger than E S ⫽13 meV (Jung, 2000). A common feature in many layered manganites, the magnetic transition and a maximum in the thermoelectric power occur well above the temperature at which the electrical resistivity peaks (Hur et al., 1998). This has been interpreted by several authors as indicative of small polaron cluster formation. Thermal-conductivity measurements were carried out in La1.2Sr1.8Mn2O7 crystals (Matsukawa et al., 2000). The thermal conductivity is anisotropic, being between three and five times larger in the ab direction. The thermal conduction in the ab direction resembles that of the cubic samples [Fig. 29(d)] with a positive slope above T C and a feature at the onset of the ferromagnetic transition. Along the c direction the thermal conductivity increases as the temperature decreases, in agreement with what one would expect for most solids as the phonon mean free path increases, with no feature at the Curie temperature. V. FERROMAGNETIC/PARAMAGNETIC PHASE TRANSITIONS A. Theoretical background The transition between the ferromagnetic state described in Sec. III.C and the high-temperature polaronRev. Mod. Phys., Vol. 73, No. 3, July 2001 like behavior discussed in Sec. IV motivated the current resurgence of interest in these materials. The extreme sensitivity of the transport properties, the resistivity in particular, to applied magnetic field is, of course, the essence of colossal magnetoresistance. As discussed in Sec. II.C, the dependence of T C on the charge-carrier bandwidth in a double exchange system, and the dependence of the bandwidth on temperature, led Kubo and Ohata (1972) to the conclusion that the nature of the phase transition would differ from the behavior expected for a 3D Heisenberg ferromagnet. More recently, Furukawa (1995a) used the ferromagnetic Kondo lattice model Hamiltonian for infinite spin and dimension, H⫽⫺t 兺 具 ij 典 , 共 c i† c j ⫹H.c.兲 ⫺J 兺i i •m i , (18) to model the double exchange system and confirmed the linear dependence of T c on the bandwidth. Within the context of the Kubo-Ohata and Kondo lattice-model approaches (Furukawa, 1994, 1995b) the resistivity below the magnetic transition decreases as 共 M 兲 ⫽ 共 M⫽0 兲共 1⫺CM 2 兲 , (19) and is therefore constant above T C . However, materials that exhibit colossal magnetoresistance are characterized by exponentially increasing resistivity as the temperature is reduced toward T c . The failure of earlier models to produce semiconductive behavior was remedied in a calculation by Millis et al. (1995) that takes into account fluctuations in the angle ij ⫽cos⫺1(Sជ i•Sជ j /S2), where i and j are neighboring sites on the lattice. The resistivity involves the correlation function 具 0 ␦ 1 R ␦ 2 典 , where ␦ 1 and ␦ 2 are nearest neighbors of sites labeled 0 and R. Millis et al. point out that this differs from the usual Fisher-Langer resistivity, which involves only 具 0 ␦ 1 典 correlations and that, because static mean-field theory ignores fluctuations, it misses the upturn in resistivity. The resistivity calculated in the dynamic mean-field model is shown in Fig. 40 along with data from Tokura et al. (1994). The calculated resistivity is several orders of magnitude smaller than observed and requires an unreasonably large field to produce significant magnetoresistance. The conclusion is that a major element is missing from the model, namely, the JahnTeller interaction that characterizes the parent compound. In fact, each Mn4⫹ site, lacking an e g electron, does not gain energy from a Jahn-Teller distortion of its local environment. It is likely, then, that hole motion is correlated with an ‘‘anti-Jahn-Teller’’ distortion, by which we mean an empty e g site that is undistorted relative to neighboring occupied e g orbitals and that takes the form of the polarons discussed in Sec. IV. The effect of Jahn-Teller coupling was first treated by Röder, Zang, and Bishop (1996), who added it to Eq. (18) and demonstrated the tendency of carriers to become self-trapped once the effective Jahn-Teller coupling is sufficiently large. A subsequent calculation by Millis and co-workers (Millis, Shraiman, and Mueller, 1996; Millis, Mueller, and Shraiman, 1996) reproduced 614 M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 40. Resistivity calculated from the double exchange model without electron-phonon interaction: solid line, the resistivity in zero field; dashed line, the resistivity in a field of 0.15T C , corresponding to 200 T. The inset displays data from Tokura et al. Both the clculated absolute change in resistivity and the magnetoresistance are much smaller than experiment. From Millis et al., 1995. the qualitative behavior observed in manganites near optimal doping. Figure 41 shows the magnetoresistance in this model for values of the effective Jahn-Teller coupling constant just below (upper panel) and just above (lower panel) the critical value for self-trapping. The effective coupling parameter ⫽g 2 /tk depends not only the Jahn-Teller coupling strength g, but also inversely on the bandwidth through t and the lattice stiffness via k. As the dopant atom M changes from Ca to Pb to Sr in La2/3M 1/3MnO3 , the bandwidth increases as the MnO-Mn bond angle tends toward 180°. Concomitantly, decreases such that Ca- and Pb-doped samples are above the critical value and Sr-doped samples, below. Substitution of Nd for La also affects the bond angle, and makes the system more prone to polaron formation. Typically, tabulated values of ionic sizes are used to estimate the deviation of the crystal structure from ideal perovskite structure through the tolerance factor f. As in the original Kubo-Ohata calculation and as shown in Fig. 42, the transition temperature depends on , which depends on the effective hopping matrix element t which, in turn, depends on the short-range spin order. This bootstrap mechanism was treated by Millis, Mueller, and Shraiman (1996) by means of a dynamical meanfield method which yields sharp, but continuous (second-order) transitions. B. Nature of the phase transition Despite the enormous body of literature in this field, relatively little effort has focused on the nature of the Rev. Mod. Phys., Vol. 73, No. 3, July 2001 FIG. 41. The effect of including Jahn-Teller effects, characterized by an effective coupling parameter in the double exchange model. The temperature dependence of the resistivity is shown for different values of magnetic field h for (a) ⫽0.7 and (b) ⫽1.12. The parameter h is related to the physical field through h⫽g B S c H phys /t. Using g⫽2, t⫽0.6 eV, and S c ⫽3/2, we find that h⫽0.01 corresponds to H phys ⫽15 T. From Millis, Mueller, and Shraiman, 1996. phase transition beyond the obvious changes in resistivity. As the transition temperature is decreased, whether by doping or ion-size substitution, self-trapping becomes more evident through the activated behavior of the conductivity. Once T C is reduced to 200 K or so, the transition becomes hysteretic and clearly first order. This is readily seen in the data of Fig. 43 (Hwang et al., 1995), where the tolerance factor (and along with it, T C ) is reduced by substitution of Pr for La. Hysteresis on cooling and heating becomes increasingly significant as the transition temperature decreases (and the CMR effect increases). Even above that temperature, where there is no strong hysteresis, the transition differs markedly from that of a conventional Heisenberg ferromagnet. Clear evidence for this is seen in the heat-capacity data on polycrystalline La0.7Ca0.3MnO3(LCMO) by Park, Jeong, and Lee (1997). Rather than broadening with increasing field, the specific-heat peak shifts to higher temperature with little change in shape. This has been confirmed in single crystals of the same composition (T C ⫽216 K) by Lin et al. (2000) as seen in Fig. 44. The heat capacity peak is very sharp, suggesting a nearly first- M. B. Salamon and M. Jaime: Manganites: Structure and transport 615 FIG. 42. The dependence of the Curie temperature T C on the effective Jahn-Teller parameter . Because it is inversely proportional to the bandwidth, decreases as magnetic order sets in, causing the Curie temperature to increase with decreasing temperature. Millis, Mueller, and Shraiman, 1996. order transition. This contrasts with the heat-capacity peak of La0.7Sr0.3MnO3 (T C ⫽359 K) which is much more lambda like (Lin et al., 2000) and which shifts relatively little in applied fields up to 1 T. A small-angle neutron study (Ibarra et al., 1998) of a nominal La2/3Ca1/3MnO3 sample (T C ⬇250 K) also showed unusual characteristics associated with the transition. While the quasielastic line shape is Lorentzian as expected, the inverse of its width, which measures the ferromagnetic correlation length, shows no power-law increase over the range 1.02T C ⭐T⭐1.15T C . Rather, the intensity of the quasielastic peak increases as T C is approached then drops abruptly, following closely the anharmonic contribution to the volume thermal expansion (Ibarra et al., 1998). Inelastic neutron scattering on a comparable sample finds that the spin-wave stiffness coefficient D does not tend toward zero as for a conventional ferromagnet, but rather retains 50% of its low temperature value at T C [see Fig. 23(c)]. In place of the collapse of spin-wave groups toward zero energy, a strong diffusive peak appears which grows in intensity as T C is approached from below and persists to at least 1.1T C (Lynn et al., 1996). A subsequent neutronscattering study by Fernandez-Baca et al. (1998) found very similar behavior in Nd0.7Sr0.3MnO3 (T C ⫽198 K) but more conventional behavior in Pr0.63Sr0.37MnO3 (T C ⫽301 K). Relatively little attention has been paid to the critical point properties of CMR manganites. This is not surprising considering the evidence presented above that the transition is percolative in nature. As the transition temperature is increased, either by doping or by changes in tolerance factor, the tendency for polaron formation decreases and the resistive behavior is metallic over the entire temperature range. Referring to Fig. 45, we see that the innermost contour of the ferromagnetic metal regime represents the most metallic, and possibly the most ‘‘normal’’ ferromagnetic transition. Indeed, a scaling analysis of a single crystal of La0.7Sr0.3MnO3 (T C Rev. Mod. Phys., Vol. 73, No. 3, July 2001 FIG. 43. Progression of the resistivity and magnetoresistance with changes in trivalent ion: top panel; ln (T) in 0 and 5 T for a series of samples of La0.7⫺x Prx Ca0.3MnO3 (x⫽0, 0.175, 0.35, 0.525, 0.6, 0.7) and La0.7⫺y Yy Ca0.3MnO3 (y⫽0.35 and 0.5). Bottom panel, magnetoresistance (cooling data) for x⫽0, 0.175, 0.35, 0.525, 0.6, 0.7 specified as ln(0T⫺5T)/5T , with the maximum in 0T indicated by arrows. The interpretation is that increasing the Pr content increases the effective Jahn-Teller coupling constant by reducing the hopping matrix element. From Hwang et al., 1995. ⫽354 K) by Ghosh et al. (1998) gives critical exponents  ⫽0.37, ␥ ⫽1.22, and ␦ ⫽4.25, as shown by the data collapsing shown in Fig. 46. These values differ from 3D Heisenberg values by amounts outside experimental uncertainty, and must be regarded as effective exponents that reflect some residual tendency for mixed-phase behavior. As expected, samples with 20% Sr doping (T C ⬇310 K) exhibit quite different effective critical exponents that are far from Heisenberg values (Mohan et al., 1998; Schwartz, 2000). The pyrochlore CMR compound Tl2Mn2O7 , which has no indication of polaron formation, exhibits critical exponents that are very close to theoretical values for the 3D Heisenberg model (Zhao et al., 1999). These results support the idea that the critical temperature, reflecting the combined effects of JahnTeller coupling, bandwidth, and lattice stiffness, serves as a measure of the degree to which polaron formation drives the material toward a first-order phase transition. 616 M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 44. Heat capacity of La0.7Ca0.3MnO3 plotted vs temperature in external fields of 0, 10, and 70 kOe. Unlike conventional ferromagnetic transitions, the peak moves to significantly higher temperatures with applied field. The peaks do not obey the usual critical-point scaling behavior. From Lin et al., 2000. Further evidence for an unconventional phase transition in LCMO is found in muon relaxation data (Heffner et al., 1996). The muon precession frequency decreases with increasing temperature, but tends to zero at a temperature above the T C deduced from the magnetization. In addition to the precessing component, which relaxes at a rate comparable to the precession frequency, a second contribution is found that relaxes at a rate several orders of magnitude slower. Initial studies of the slow component showed a peak in the relaxation rate at the magnetization value of T C , but with nonexponential shape. Below T C the relaxation function is a stretched exponential of form exp关⫺(⌳t)1/2兴 , slowly evolving into a simple exponential by 1.2T C . The authors suggest a glassy state in which fast-relaxing and slow-relaxing components coexist for times on the order of ⬃1 s. More recently, Heffner et al. (2000) have combined neutron spin-echo and muon relaxation data to identify two distinct components to the relaxation rate in LCMO, in place of a stretched exponential. One component is identified with an extended cluster and exhibits fast spin dynamics and critical slowing down, which translates into slower muon relaxation, and an increasing volume fraction as temperature is reduced. This component is observed only at small momentum transfer (large length scales) in the spin-echo data. The second component is characterized by slowly fluctuating Mn spins (rapid muon relaxation), and a decreasing volume fraction. These results strongly suggest that the ferromagnetic Rev. Mod. Phys., Vol. 73, No. 3, July 2001 FIG. 45. Phase diagram in the plane defined by the average A-site covalent radius 具 r co v 典 and divalent ion concentration x, showing the various antiferromagnetic insulating (AFI), ferromagnetic metal (FMM), ferromagnetic insulator (FMI), and charge ordered insulating (COI) phases. Inside the region labeled FMM, contours of T C are shown. The dotted lines show the effect of Sr, Ca, and Ba substitution. The bold arrow indicates that the effective Jahn-Teller coupling, , increases as T C decreases. Charge order is most stable near x⫽0.5 and x ⫽0.6– 0.67. From Ramirez, 1997. transition proceeds by means of changing volume fractions of rapidly and slowly relaxing local magnetization. C. Two-phase behavior An early suggestion that two-phase separation might govern the colossal magnetoresistive regime was made by Gor’kov (1998). Monte Carlo simulations of the double-exchange model with Jahn-Teller coupling also suggested phase segregation of ferromagnetic metal from antiferromagnetic insulator regimes. It is necessary to make a distinction here between phase separation, by which is meant coexistence of ferromagnetic/conducting and paramagnetic/insulating regions with constant carrier density, and charge segregation as suggested by Yunoki, Moreo, and Dagotto (1998). An experimental test of these ideas was carried out by Jaime et al. (1999) using an effective-medium approach. The resistivity data were analyzed in terms of a temperature and magneticfield-dependent concentration c(H,T) of the metallic phase. The power-law resistivity of the metallic regime and the activated conductivity of the polaronic regimes were assumed to persist into the mixed phase regime near the metal-insulator transition. Combining these with the measured resistivity, Jaime et al. extracted the metallic concentration using data from a thin-film sample of La2/3Ca1/3MnO3 using effective-medium expressions. Because the film thermal conductivity is dominated by that of the substrate, the thermal conductivity can be eliminated from the transport equation to give a very simple expression for the thermoelectric power, M. B. Salamon and M. Jaime: Manganites: Structure and transport 617 FIG. 46. Critical-scaling plots of La0.7Sr0.3MnO3 below and above the Curie temperature (T C ⫽354.0 K), using  and ␥ as noted in the text. Inset shows the same plot on a log scale. ⑀ is equal to 兩 T⫺T C 兩 /T C . From Ghosh et al., 1998. S 共 H,T 兲 ⫽ 1 关 S ⫺ met S pol pol ⫺ met pol met ⫹ exp共 H,T 兲共 S pol ⫺S met 兲兴 , (20) where the subscripts refer to polaron, metallic, and experimental values. As with the resistivity, the thermoelectric powers are fit in the two regimes and assumed to characterize the two phase region. The extracted values of c(H,T) and the fits to the thermopower are shown in Fig. 47. A number of other experimental probes provide further evidence for the mixed nature of the transition in the CMR regime. EXAFS studies of a series of La1⫺x Cax MnO3 samples, with x⫽0.21, 0.25, and 0.30 were performed by Booth et al. (1998). The meansquare width of the Mn-O bond-length distribution decreases as the temperature is reduced, approaching at low temperatures the values for x⫽1, for which there is no Jahn-Teller distortion. Similar results were found from neutron pair-distribution-function data for La0.8Sr0.2MnO3 by Louca and Egami (1999). As seen in Fig. 48, the number of short Mn-O bond lengths changes from approximately four (indicating polaronic elongation of the oxygen octahedra) at high temperatures toward six (no average distortion) at low temperatures. Similar results for La1⫺x Cax MnO3 have been reported by Billinge et al. in which the two-phase coexistence reRev. Mod. Phys., Vol. 73, No. 3, July 2001 FIG. 47. Two-phase analysis of resistivity and thermopower: (a) The concentration of metallic regions c(H,T) extracted from the resistivity data using the effective-medium approximation and assuming elongated conducting regions with length-to-width ratio a/b⫽50; (b) Seebeck coefficient data and results of a computation using the measured resistivity exp(H,T) and an effective-medium approximation as described in the text. Dotted lines show the low- and hightemperature fits. From Jaime et al., 1999. gion is mapped out via pair-distribution-function analysis (Billinge et al., 2000). Particularly convincing evidence of two-phase coexistence is found in the Raman data of Yoon (1998). The spectrum for a sample of Pr0.7Pb0.21Ca0.09MnO3 exhibits, above its Curie temperature T C ⫽145 K, two features in addition to phonons: a broad peak centered near 1100 cm⫺1 and a lowfrequency diffusive response. The peak is attributed to the photoionization energy of small polarons and the diffusive signal, to their hopping motion. As the temperature is reduced, the diffusive signal evolves into a flat continuum response characteristic of metallic behavior. Figure 49 shows the temperature dependence of the integrated Raman intensities for the diffusive (polaronic) and continuum (electronic) contributions. Their coexistence in the vicinity of T C is readily apparent. Quite recently, diffuse neutron-scattering peaks have been detected (Dai, Fernandez-Baca et al., 2000) that indicate polaron-polaron correlations exist over a wide 618 M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 48. Bond-length distribution: Top, the number N Mn-O of short Mn-O bonds as a function of temperature for La0.8Sr0.2MnO3, as determined by neutron pair-distributionfunction analysis. The presence of four short bonds at high temperatures is indicative of the existence of Jahn-Teller polarons. The bond lengths tend to equalize with decreasing temperature, but some residual distortion of the oxygen octahedra remains at low temperatures. Bottom, the height of the pairdistribution function peak at 2.75 Å which includes the O-O pair. Its relation to temperature is similar to that of the N Mn-O value. From Louca and Egami, 1999. temperature range above T C , but collapse below. The position of the peak suggests short-range charge ordering with a correlation length of ⬃1.4 nm at room temperature, increasing to ⬃2.8 nm at the transition. There is no sign of long-range charge ordering. Further evidence for mixed phases comes from a Mossbauer study of LCMO doped with 57Co (Chechersky et al., 2000). Coexisting ferromagnetic and paramagnetic sites are found in the vicinity of the phase transition, as might be expected from the conducting/insulating mixture postulated by Jaime et al. (1999). Additional evidence for inhomogeneous conductivity in the vicinity of the phase transition is found on the noise characteristics of film and single-crystal samples. Early work (Hardner et al., 1997) on a partially annealed La2/3Ca1/3MnO3 film (T C ⬇100 K) revealed presence of significant non-Gaussian noise in the form of discrete resistance switching that is largely absent well below the transition, but predominant at T C . Anomalously large 1/f noise is also found in LCMO films (Alers et al., 1996; Raquet et al., 2000), and it has been found that the magnitude in epitaxial films is sensitive to oxygen content (and therefore T C ) (Rajeswari et al., 1998). Samples that are in the first-order transition regime show very large 1/f noise which is taken as evidence for a percolation transition (Podzorov et al., 2000). Recent noise measurements on single crystals also finds a large 1/f contribution in LCMO with evidence for two-state switching in a narrow temperature range near T C ⫽218 K. From the field and temperature characteristics of the switchers, the authors conclude that regions as large as 104 unit cells are fluctuating between high and low resistance states, providing strong evidence for inhomogeneous conducting paths as expected for a percolationlike transition (Merithew et al., 2000). When the transition is fully first-order in character, the phase separated regions are mesoscopic (micrometer range) and have been observed by electron microscopy (Uehara et al., 1999). Static images of conducting and insulating regions have also been captured using scanning-tunneling microscopy (Tath et al., 1999). In this study, coexisting insulating and conducting regions on the 100-nm scale were imaged, and the growth of metallic regions was monitored as a magnetic field was applied near the transition temperature. Such quasistatic objects may be related to the slow-switchers observed in noise studies, but are larger in scale that those detected by neutron scattering. A recent calculation has indicated that these giant clusters have equal electronic density and result from disorder in the exchange and hopping amplitudes in the vicinity of a first-order transition (Moreo et al., 2000). D. Phase-separation models FIG. 49. Raman-scattering data showing the temperature dependence of the integrated scattering strength ratio for 䊉, the diffusive response, I diff (T)/I diff (T⫽350 K) indicative of polarons, and 䊏, the flat continuum response, I cont (T)/I cont (T⫽90 K) indicative of band electrons, for a T C ⫽145 K sample. From Yoon et al., 1998. Rev. Mod. Phys., Vol. 73, No. 3, July 2001 The effective-medium approach suggests that metallic and insulating regions coexist as interpenetrating clusters, also suggesting a percolation picture of the insulator-metal transition. One such percolative model was proposed by Bastiaansen and Knops (1998) based on a random resistor network. A Monte Carlo simulation of a 2D Ising model formed the basis of the calculation, with unit resistors connecting aligned nearestand next-nearest-neighbor sites and infinite resistance linking unaligned sites. The resistance of the network is qualitatively similar to experiment. For a system of Heisenberg spins, of course, resistance of each link can vary continuously between 1 (linked spins parallel) and M. B. Salamon and M. Jaime: Manganites: Structure and transport 619 FIG. 51. Inverse magnetic susceptibility H/m vs temperature near the M-I transition for H⫽24 kOe. Dashed line, the zerofield extrapolated behavior. The appearance of free carriers induces an increase in the effective T C , leading to a kink in the susceptibility. Inset: inverse susceptibility of a single-crystal sample of La0.7Ca0.3MnO3, measured in a field of 1 kG. From Jaime et al., 1999. Teller coupling constant eff is smaller than the critical value c . Because eff is, in turn, inversely proportional to the bandwidth, the double exchange mechanism causes eff to decrease as magnetic order sets in. In the CMR regime, where eff is presumed to be near c , it can be expanded as FIG. 50. Results of a coupled-order parameter model: (a) the conducting concentration c(H,T) calculated in the mean-field model with ␣ ⫽0.02 and ␥ ⫽0.3; (b) the magnetization calculated with the same parameters. The dotted line shows the noninteractive case for comparison. From Jaime and Salamon, 1999. infinity (linked spins antiparallel), greatly reducing the size of the resistance peak just at T C . Quite similar results have been reported recently by Mayr et al. (2001). A more realistic model was proposed by Lyukyutov and Pokrovsky (1999) which is based on Varma’s theory (Varma, 1996) of magnetic polaron formation, modified to include Jahn-Teller effects. Magnetic polarons, which coexist with small lattice polarons, are assumed to be large, basically comprising magnetically correlated regions. As the temperature is lowered, the magnetic polaron density increases until the magnetic polarons overlap, which defines the percolation point. The authors argue that long-range Coulomb effects render implausible suggestions that macroscopic charge separation underlies the CMR effect (Yunoki et al., 1998; Moreo et al., 1999). Similar ideas have been discussed by Gor’kov (Gor’kov, 1998; Gor’kov and Kresin, 1998). A simple mean-field model has been proposed by Jaime et al. (1999) that gives a qualitative description of the conducting fraction c(H,T). A simplification of the dynamic mean-field theory of Millis, Mueller, and Shraiman (1996), it recognizes that the fully metallic state c(0,T→0) is achieved only if the effective JahnRev. Mod. Phys., Vol. 73, No. 3, July 2001 eff⯝ c ⫹ ␣ ⫺ ␥ m 2 , (21) where ␣ is a small positive constant that assures that polarons will form once the magnetization m ⫽M(H,T)/M sat becomes smaller than ␣ / ␥ . The conducting fraction is then taken to be an order parameter satisfying c⫽tanh关共 1⫺ ␣ ⫹ ␥ m 2 兲 c 兴 . (22) The coupling constant ␥ causes the Curie temperature to increase when c is finite, so that the magnetization in zero field rises more rapidly for c⭓0. Figure 50 shows the conducting fraction and magnetization at several magnetic fields for ␣ ⫽0.02 and ␥ ⫽0.4, which compare well with c(H,T) seen in Fig. 47(a). There is a kink in the magnetization at the point at which the mixing factor begins to increase; such kinks are frequently observed experimentally as seen in Fig. 51 (Jaime et al., 1999). This model cannot, of course, capture the essential contributions of magnetic/conducting fluctuations. E. Hall effect in the transition region Even in the regime where the charge carriers are nominally ‘‘metallic,’’ estimates based on free-electron ideas (electron density n⬇1027 m⫺3, effective mass m⬇3m e ) suggest that the mean free path approaches interatomic distances once the resistivity exceeds 1 m⍀ cm. Therefore even the metallic portions of any percolation network must be considered to be localized 620 M. B. Salamon and M. Jaime: Manganites: Structure and transport pend only on the magnetization, independent of the details of the hopping process, and is given by 0 xy ⫽ xy 冉 冊 m 共 1⫺m 2 兲 2 . 1⫹m 2 (23) Figure 52 shows that this is obeyed by the data of Fig. 0 that are in reasonable agree27(a) with values of xy ment with estimates using atomic spin-orbit coupling constants. The boundary between this anomalous contribution and the ordinary (Holstein) contribution was found recently by Chun, Salamon, Jaime et al. (2000) as a minimum in the Hall mobility. VI. CHARGE AND ORBITAL ORDERING TRANSITIONS A. 3D manganites FIG. 52. Plot of the Hall resistivity xy vs reduced magnetization using the data shown in Fig. 27 showing scaling behavior: solid line, a fit to Eq. (23); dashed line, the numerator of Eq. (23) only. There are no fitting parameters except the amplitude. From Chun, Salamon, Lyanda-Geller et al., 2000. (Gor’kov, 1998; Lyanda-Geller et al., 2000). The degree to which the localization length shrinks to permit selftrapping by Jahn-Teller processes determines the strength of the CMR effect. However, once the resistivity exceeds this Mott-Ioffe-Regel limit, all transport processes must be considered within a hopping framework. This point is made forcefully by the strongly anomalous behavior of the Hall effect which is very similar for a variety of manganites despite major differences in their resistive properties. Figure 27(a) shows the Hall resistivity of the single-crystal sample whose longitudinal resistivity was shown in Fig. 1. Despite the very large differences in xx among various samples, the xy data are remarkably similar, and deviate strongly from the behavior of conventional ferromagnets in the critical region. In conventional magnets, Eq. (13) permits the separation of ordinary and extraordinary components through the critical point, here the Hall resistivity becomes nonmonotonic and strongly curved. Only at high temperatures, where the polaronic effects dominate, and at very low temperatures, can an ordinary Hall coefficient be defined, as seen in Figs. 27(a) and (b). The first suggestion that a mechanism beside skew scattering and sidejump processes was needed was made by Ye et al. (1999). Formulated for metallic conduction, this model predicted a maximum in xy above the Curie temperature, which is contrary to what is observed. Recognizing that the entire transition region is in the nonmetallic, hopping regime, Chun, Salamon, Lyanda-Geller et al. (2000) extended the classic Hall calculation of Holstein (1959) for the Hall constant of hopping charge carriers to take account of the Berry-like phase that is accumulated by charge carriers that are required by strong Hund’s-rule coupling to follow the local spin texture. The model predicts that the Hall resistivity should deRev. Mod. Phys., Vol. 73, No. 3, July 2001 Away from the ferromagnetic regime, and even overlapping it somewhat, is a variety of charge and orbitally ordered phases. The experimental situation has been reviewed in some detail recently by Rao and Raveau (Rao et al., 2000). It might be argued that charge ordering at doping levels that are rational fractions represent an order-disorder transition of Jahn-Teller polarons. This interpretation was favored by Cheong and Chen (1998) who found evidence for charge ordering in La1/2Ca1/2MnO3 and La1/3Ca2/3MnO3 . Indeed, looking at Fig. 12 we would expect that approach to describe Sm1/2Ca1/2MnO3 , Nd1/2Ca1/2MnO3 , and Pr1/2Ca1/2MnO3 and, for the last of these, direct evidence for it from high-resolution electron microscopy has been reported (Li et al., 1999). However, this picture cannot hold for Nd1/2Sr1/2MnO3 or Pr1/2Sr1/2MnO3 , where the chargeordered phase arises within the more metallic, ferromagnetic regime. Figure 53 shows the contrast between Sm1/2Ca1/2MnO3 and Nd1/2Sr1/2MnO3 (Tomioka et al., 1997). While the former shows a weak peak in magnetization and an insulator/insulator transition at the charge ordering temperature, the latter clearly becomes ferromagnetic and metallic near 250 K before undergoing a first-order metal/insulator transition to an antiferromagnetic phase near 160 K. That Nd1/2Sr1/2MnO3 and Pr1/2Sr1/2MnO3 both become antiferromagnetic at the charge ordering transition is shown by the neutrondiffraction data (Kawano et al., 1997) in Fig. 54. That figure also demonstrates the narrowness of the antiferromagnetic CE phase (see Fig. 11 for its structure) shown in Fig. 17. This has been explored in some detail by Kajimoto et al. (1999). At a Sr content of x⫽0.55 the ferromagnetic phase does not intercede between the paramagnetic state and the insulating, A-type antiferromagnetic phase. As was noted previously, the tendency toward stabilization of Jahn-Teller polarons is strongly correlated with the ionic size of the divalent dopant in the A site. This has been attributed to the intrinsic bandwidth to be expected in the fully metallic state, which in turn depends on the Mn-O-Mn bond angle. At x⫽1/2, as well, the primacy of the tendency toward charge ordering and an insulating state is closely tied to the tolerance M. B. Salamon and M. Jaime: Manganites: Structure and transport 621 FIG. 53. Charge ordering transitions in Sm1/2Ca1/2MnO3 and Nd1/2Sr1/2MnO3. NSMO exhibits a ferromagnetic, metallic phase before undergoing a charge ordering transition. In contrast, SCMO makes a transition directly to the insulating, charge ordered state. From Tomioka et al., 1997. FIG. 54. Neutron-scattering results on the ferromagnetic-toantiferromagnetic transitions in x⫽1/2 manganites showing A-type (Pr1/2Sr1/2MnO3) and CE-type (Nd1/2Sr1/2MnO3) order. The latter result on NSMO supports the data shown in Fig. 53. Off x⫽1/2 the ferromagnetic phase is absent and A-type order occurs. From Kawano et al., 1997. Rev. Mod. Phys., Vol. 73, No. 3, July 2001 factor. Figure 55 summarizes the sequence of transitions for various combinations of A-site atoms at x⫽1/2. A dramatic feature of the charge ordered state is that it can be ‘‘melted’’ by a magnetic field, as first reported by Kuwahara et al. (1995). The transition, which is hysteretic in temperature at zero field, becomes strongly hysteretic in applied fields, as may be seen in Fig. 12. The melting transition is clearly observed in the resistivity, as is shown in Fig. 56 for Pr1/2Ca1/2MnO3 . The changes are even more dramatic (Tokura et al., 1996) for Nd1/2Sr1/2MnO3 , as seen in Fig. 57(a). At low temperatures, there is a resistive change of six orders of magnitude in a field of 7 T. Kuwahara and co-workers (Kuwahara et al., 1995; Kuwahara and Tokura, 1998) have mapped out the field-temperature phase diagram for this compound by cycling the field at fixed temperature. As seen in Fig. 57(b), the region of bistability grows dramatically as the temperature is reduced, with the metallic state tending toward metastability. Kuwahara and Tokura (1998) suggest that there exist two local minima in the free energy of the system, one corresponding to a large-M, charge liquid state and the other a small-M, charge ordered one, separated by a barrier U. At a critical value of the field, the Zeeman energy favors the metastable, large-M state and the system becomes metallic. As the field is removed, the barrier prevents the return of the system to its stable state, tending to trap it in the metastable, conducting state. When the hole concentration is not exactly 1/2 or 2/3 and, in the case of La1/2Ca1/2MnO3 , possibly even at x ⫽1/2 (Cheong and Chen, 1998), charge order is found to be incommensurate. There is evidence for coexistence of the x⫽1/2 and 2/3 structures as well as discommensurations. As a consequence of the discommensurations, there is a spin slip associated with the Mn3⫹ sites that results in a shorter correlation length for Mn3⫹ sublattice (Cheong and Chen, 1998). Figure 58 is a schematic 622 M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 55. Dependence of charge-ordering transition on ionic size or tolerance factor for various manganites. Phases include paramagnetic insulator (PI), a charge-ordered insulator (COI), and a ferromagnetic metal (FM), respectively. From Kuwahara and Tokura, 1998. sketch of a discommensuration that results from a spin (and orbital) slip along a line of Mn4⫹ ions. There have been several theoretical efforts to understand the charge ordering mechanism and the emergence of the CE antiferromagnetic state. A general study of charge ordering in the context of the Holstein model was carried out by Chiuchi and de Pasquale using dynamical mean-field theory (Chiuchi and de Pasquale, 1999). They identify a phase boundary between a randomly distorted, charge ordered state and a polaronic ordered phase as a function of the effective electronphonon coupling parameter (see Sec. IV). van Veenendal and Fedro argue that when polarons encompass two sites, the hopping of the electron between them FIG. 57. Field-induced metastability at x⫽( 12 ); (a) fieldinduced metal-insulator transition in Nd1/2Sr1/2MnO3 taken from a series of Sm-doped samples From Tokura et al., 1996; (b) field-temperature phase diagram for Nd1/2Sr1/2MnO3. The hatched area corresponds to a region of metastability. From Kuwahara et al., 1995. ‘‘dresses’’ the polaron, and provides an explanation for the ferromagnet-CE phase transition (van Veenendaal and Fedro, 1999). Shen and Wang (1999) demonstrate that Wigner crystallization occurs within the double exchange model when processes that allow the possibility of S-1/2 states, where S is the spin of the t 2g core, even absent Jahn-Teller coupling. This may be relevant to the Wigner crystal ordering observed at 2/3 filling. B. Layered manganites FIG. 56. Field-induced metal-insulator Pr1/2Ca1/2MnO3. From Tomioka et al., 1996. Rev. Mod. Phys., Vol. 73, No. 3, July 2001 transition in As discussed in Sec. III.B.2, the bilayer manganites RESr2Mn2O7 also show charge ordering effects. The situation is more complicated with the charge-ordered state exhibiting a competition between A-type and CE-type magnetic order. In a study of (Nd1⫺x Lax )Sr2Mn2O7, Moritomo et al. (1999) found a M. B. Salamon and M. Jaime: Manganites: Structure and transport FIG. 58. Sketch of a discommensuration exhibiting a spin slip in the Mn3⫹ sublattice. From Cheong and Chen, 1998. narrow, charge-order CE antiferromagnetic regime to exist on the La-rich end of the series, but to dominate in a narrow temperature range between 110 and 220 K. On cooling, the resistivity of LaSr2Mn2O7 first increases sharply at the charge ordering transition, reaches a maximum, and then decreases to a minimum near 100 K. Single-crystal neutron-scattering data provide greater detail (Kubota, Fujioka et al., 1999; Kubota, Yoshizawa et al., 1999). As shown in Fig. 59, A-type antiferromagnetic order [Fig. 59(a)] and CE-type charge order [Fig. 59(b)] appear simultaneously at 210 K. This coincides with a decrease in diffuse magnetic scattering [Fig. 59(c)]. Near 120 K, the CE-type charge ordering disappears and a weak, CE-type antiferromagnetic peak [Fig. 59(d)] appears. A small volume fraction of CE antiferromagnetic order persists to low temperature, but the FIG. 59. Neutron-diffraction results for NdSr2Mn2O7. (a) Antiferromagnetism and (b) charge ordering appear at 210 K, while (c) diffuse magnetic scattering decreases below that temperature. (d). Near 120 K, the antiferromagnetic order changes to CE type over a narrow temperature range. From Kubota, Fujioka et al., 1999b. Rev. Mod. Phys., Vol. 73, No. 3, July 2001 623 main contribution disappears at the temperature at which CE charge order vanishes. The authors conclude that this ‘‘melting’’ of the charge ordered state reflects the competition between A-type d x 2 ⫺y 2 and CE-type d 3x 2 ⫺r 2 and d 3y 2 ⫺r 2 orbitals. Substitution of Nd for La lowers and reduces the resistivity peak, but leaves some remanence of the charge ordered state in the temperature range between 100 and 200 K. The resistivity of the single-layered compound La1/2Sr3/2MnO4 increases slowly with decreasing temperature to approximately 240 K, below which it increases rapidly (Moritomo et al., 1995). Electron diffraction showed this temperature to mark the onset of charge ordering. Subsequent work by Bao et al. (1996) found the charge order superlattice spots to be consistent with 3D ordering, and to lie at 1/4 positions. However, neutron-scattering data (Sternlieb et al., 1996) showed that charge order to set in near 220 K in the pattern shown in Fig. 14, with spin ordering occurring at 110 K. Peaks at the 1/4 position were observed by x-ray diffraction at the Mn K edge, and were attributed to orbital ordering in the pattern shown in Fig. 14 (Murakami, Kawada et al., 1998). VII. IMPLICATIONS Between the time of early experiments (Jonker, and van Santen, 1950; van Santen and Jonker, 1950; Volger, 1954) and subsequent theoretical treatments (Zener, 1951; Anderson and Hasegawa, 1955; Goodenough, 1955; de Gennes, 1960) in the 1950s and the work of Jin and co-workers in the 1990s (Jin, McCormack et al., 1994; Jin, Tiefely et al., 1994) this field lay dormant. The double exchange mechanism was partly at fault, lulling researchers into believing that it contained the essential elements needed to understand these materials. The reawakening of the field came as a consequence of the large increases in magnetoresistance discovered in ferromagnetic/nonmagnetic metallic multilayers, and their ability to increase dramatically the areal bit density in magnetic recording. However, we have seen the CMR effect is only one manifestation of a competition between double exchange, that favors ferromagnetic order, and a combination of Jahn-Teller coupling, Coulomb interactions, and antiferromagnetic superexchange, which favor various flavors of charge/orbital/antiferromagnetic order. The main impact of the double exchange mechanism is to serve as an amplifier of the magnetic field in that alignment of neighboring spins serves as a valve that controls the flow of doped-in holes through the double exchange transfer matrix element t eff ij ⫽t cos(/2). The amplification can be readily seen in the heat capacity of La0.7Ca0.3MnO3 near the ferromagnetic transition, Fig. 44 where a field of 7 T, corresponding to 12 K for S⫽2, gives rise to a shift in the transition temperature more than three times larger. This even more dramatically demonstrated in Fig. 57(a), where a 7-T field drives the charge ordering transition from 150 K to zero, a tenfold amplification of the field energy. Here again, aligning spins delocalizes the doped-in hole, re- 624 M. B. Salamon and M. Jaime: Manganites: Structure and transport duces its kinetic energy, and in turn reinforces the effect of the field. In that way, a relatively small magnetic field creates a band and with it, an energy gain of 3/5 of the Fermi energy which, at half filling, is 3/10 of the bandwidth. Taking the bandwidth to be of order (104 K)k B (Coey et al., 1999) this yields more than the 150 K required to balance the charge ordering energy of Nd1/2Sr1/2MnO3. The amplification of magnetic order described above tends to overcome the tendency for charge/orbital ordering and to drive, thereby, the colossal magnetoresistance. How this comes about has been clarified by recent neutron-scattering (Dai, Fernandez-Baca et al., 2000) and electron-diffraction results (Zuo and Tao, 2001), on La2/3Ca1/3MnO3 . Diffuse, nonmagnetic, quasielastic peaks are observed in neutron scattering at (1/4, 1/4, 0) positions, indicating the presence of short-range (1–2 nm) CE-type antiferromagnetic orbital ordering. Confirming data from electron diffraction reveals the presence of 1/2-order diffuse charge peaks of comparable correlation length. The CE-type diffuse intensity increases with decreasing temperature, reaching a maximum near the temperature of the resistivity maximum. At the same time, underlying diffuse scattering attributed to uncorrelated polarons, relatively constant above T C , drops gradually below. To achieve such large volumes of fully charge ordered regions with alternating Mn3⫹ and Mn4⫹ sites in a sample with only 1/3 of the sites doped to Mn4⫹ would require excessive Coulomb energy. Rather, there must be slightly hole-rich regions, exhibiting CE-type antiferromagnetic charge modulations, surrounded by hole-poor regions. Because CEtype antiferromagnetic charge order is stripelike, the manganites in this regime resemble, to some extent, the ‘‘electronic liquid crystal phases’’ that have been proposed for cuprates doped to comparable levels. As the temperature is reduced, the hole-rich regions find themselves above the global Curie temperature, and develop ferromagnetic correlations. These correlations then suppress the tendency toward charge/orbital order, cause the diffuse CE-type peaks to lose intensity, and produce local conducting regions, exactly as postulated by the two-fluid model. This picture is also consistent with the observation that ferromagnetic correlations do not grow critically as the phase transition is approached, but rather saturate at roughly the same 1–2-nm scale observed for the size of CE-correlated regions. The competition between charge/orbital ordering and ferromagnetism raises fundamental questions about the nature of the transitions. Away from the ferromagnetic concentration regime, the charge ordering transitions are clearly first order in both field and temperature. However, the ferromagnetic transitions in the ‘‘nominal’’ concentration regime are not obviously first order, as is suggested by Fig. 45. However, as the heat-capacity data show, they do not behave as expected for conventional second-order transitions, either. As the bandwidth/band filling point in Fig. 45 moves into the 360-K contour, the nature of the phase transition becomes much closer to that of a conventional ferromagnetic transition, as seen Rev. Mod. Phys., Vol. 73, No. 3, July 2001 in the scaling curve of Fig. 46. This suggests that the tendency to localize charge carriers into insulating and ferromagnetic regimes causes the material to act like a ferromagnet with quenched in disorder, and therefore opens the possibility that the critical behavior is influenced by Griffiths singularities (Griffiths, 1969; Bray, 1987). Indeed, the spin dynamics in the region between the observed ordering temperature (⬇250 K for La2/3Ca1/3MnO3) and the highest temperature when all double exchange bonds are active (⬇360 K for La2/3Sr1/3MnO3) have been found to exhibit the nonexponential relaxation (Heffner et al., 1997, 2000) expected for Griffiths phases. As deduced from the resistivity, the magnetic transition occurs as regions with an excess of active bonds percolate to develop long-range order. Important theoretical challenges also remain in understanding the low-temperature properties. Despite differences in Curie temperature and doping, the underlying exchange interactions in the ferromagnetic state appear to be robust, at least as revealed by the behavior of longwavelength ferromagnetic spin waves seen in Fig. 14. It remains to be understood how the exchange interaction can be insensitive to the number of possible Mn3⫹-Mn4⫹ pairs. As pointed out in Sec. III.C.2, the resistivity in the ferromagnetic regime loses its temperature dependence at low temperatures, in violation of Mathiesson’s rule. Weak temperature dependence is expected in a halfmetallic ferromagnet, due to the suppression of spin-flip scattering. However, the very excitation of spin waves eventually restores a down-spin band and with it, the possibility of spin-flip scattering. Just how the down-spin band develops and when spin-wave scattering returns remains an open question, although a first attempt to address this has been made (Furukawa, 2000). The nature of the orbital degrees of freedom in the ferromagnetic regime also remains an open question. It has generally be thought of (Tokura and Nagaosa, 2000) as an ‘‘orbital liquid,’’ but there have been some recent suggestions that a chiral orbital liquid might be possible (Maezono and Nagaosa, 2000). From a condensed-matter perspective the tendency for stripe order to occur in both electron doped (e.g., Ca-rich) and hole-doped (e.g., La-rich) ends of the phase diagram make these materials interesting testing grounds for understanding the nature of liquid-crystallike phases in dilute electronic systems. The strong core magnetism and its partner, strong Hund-rule coupling, prevent the itinerant electrons from developing superconducting fluctuations such as is the case in cuprates. However, these materials represent one end of a continuum of transition-metal oxides with perovskite-type structures. All of them exhibit competition among charge, magnetic, and orbital order. When these are present, of equal strength, but not strong, as in the copper end point, superconductivity can occur. The manganites, along with cobaltates and nickelates, and probably ruthenates, must be fully understood before the unusual metallic behavior of the cuprates can be comprehended. It may well be that the metallic state of all these oxides M. B. Salamon and M. Jaime: Manganites: Structure and transport represents a new, non-Fermi-liquid state of dense electron systems and that new theory of metals will be required. As noted above, the initial motivation for reexamining this materials was to explore their possible applications to magnetic recording, sensing, and memory. The half-metallic ground state suggested that very large magnetoresistance could be attained by fabricating tunnel junctions. These efforts have been reviewed elsewhere. (Ramesh et al., 1998). In a similar regard, there are numerous reviews whose focus has been on the role of crystal chemistry and the microstructure of films grown by various methods. Our focus here has been on the variety of possible states and the transitions among them. Whether or not these materials eventually find useful applications, they have already expanded our understanding of what is a metal or an insulator, and what characterizes the transitions between these states. The lesson is that strong, but evenly balanced, interactions lead to dramatic changes in physical properties and that an understanding of how to treat such situations remains to be found. ACKNOWLEDGMENTS Work described in this paper was carried out in part with the support of the Department of Energy through the F. Seitz Materials Research Laboratory, Grant No. DEFG02-91ER45439. 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